Rintamaki Replication

Replication

Recall Rintamäki et al. 2017 in which the authors investigate whether renewable energy generation impacts the volatility of electricity prices in Germany and Denmark (two zones in DK). They use the standard deviation of hourly prices to measure daily price volatility (and SD of daily prices for weekly volatility). They then estimate a SARMA model in each case accouning for short-term price volatility as well as weekly seasonality in the data.

Abstract: “Although variable renewable energy (VRE) technologies with zero marginal costs decrease electricity prices, the literature is inconclusive about how the resulting shift in the supply curves impacts price volatility. Because the flexibility to respond to high peak and low off-peak prices is crucial for demand-response applications and may compensate for the losses of conventional generators caused by lower average prices, there is a need to understand how the penetration of VRE affects volatility. In this paper, we build distributed lag models with Danish and German data to estimate the impact of VRE generation on electricity price volatility. We find that in Denmark wind power decreases the daily volatility of prices by flattening the hourly price profile, but in Germany it increases the volatility because it has a stronger impact on off-peak prices. Our analysis suggests that access to flexible generation capacity and wind power generation patterns contribute to these differing impacts. Meanwhile, solar power decreases price volatility in Germany. By contrast, the weekly volatility of prices increases in both areas due to the intermittency of VRE. Thus, policy measures for facilitating the integration of VRE should be tailored to such region-specific patterns.”

In each “Replicate” tab I first replicate the model from Rintamäki et al directly (excluding the control for natural gas prices as this data is proprietary/unavailable) and then make a first attempt at measuring the volatility of hourly prices directly using gets. In the “Extending to Europe” tab I provide plots of the time series of electricity price data available from EMBER (more detail on data in that section). I have chosen to include the code blocks for your reference.

Replicate Daily

In the specification looking at daily price volatility the dependent variable of interest is the standard deviation of hourly prices per day. We have the advantage of being able to model the volatility of hourly prices directly with gets.

Models as in Rintamäki et al.

The below specifications replicate the results reported in Rintamaki et al for daily price volatility. Unfortunately, gas spot prices are not provided in their replication data as they are proprietary (available via Bloomberg portal) - those specs have been commented out below.

Pulled from replication data available at: https://doi-org.ezproxy-prd.bodleian.ox.ac.uk/10.1016/j.eneco.2016.12.019

# Required including "here("code/rintamaki_replication/data/" to data read commands in the listed files 
source(here("code/rintamaki_replication/models/read_daily_data.R"))

daily_forms <- list(
  # SPECS FOR DK1
  list(vol_dk1, wind_dk1, 2, 1, 2, 1, 7),
  list(vol_dk1, wind_pen_dk1, 2, 1, 2, 1, 7),
  list(vol_dk1, cbind(wind_dk1, exim_op1_dk1, exim_op2_dk1, exim_p_dk1), 2, 1, 2, 1, 7),
  list(vol_dk1, cbind(wind_dk1, exim_op1_dk1, exim_p_dk1), 2, 1, 2, 1, 7),
  list(vol_dk1, cbind(wind_pen_dk1, exim_op1_dk1, exim_p_dk1), 2, 1, 2, 1, 7),
  #list(fwd1(vol_dk1), list(fwd1(wind_dk1), diff(gas_dk1))),
  #SPECS FOR DK2
  list(vol_dk2, wind_dk2, 2, 1, 2, 1, 7),
  list(vol_dk2, wind_pen_dk2, 2, 1, 2, 1, 7),
  list(vol_dk2, cbind(wind_dk2, exim_op1_dk2, exim_op2_dk2, exim_p_dk2), 2, 1, 2, 1, 7),
  list(vol_dk2, cbind(wind_dk2, exim_op1_dk2, exim_p_dk2), 2, 1, 2, 1, 7),
  list(vol_dk2, cbind(wind_pen_dk2, exim_op1_dk2, exim_p_dk2), 2, 1, 2, 1, 7),
  #list(fwd1(vol_dk2), list(fwd1(wind_dk2), diff(gas_dk2))),
  # SPECS FOR DE
  list(vol_de, wind_de, 2, 1, 2, 1, 7),
  list(fwd1(vol_de), diff(solar_de), 2, 1, 2, 1, 7),
  list(fwd1(vol_de), cbind(fwd1(wind_de), diff(solar_de)), 2, 1, 2, 1, 7),
  list(vol_de, res_de, 2, 1, 2, 1, 7),
  list(vol_de, wind_pen_de, 2, 1, 2, 1, 7),
  list(fwd1(vol_de), diff(solar_pen_de), 2, 1, 2, 1, 7),
  list(vol_de, res_pen_de, 2, 1, 2, 1, 7),
  list(fwd1(vol_de), cbind(fwd1(wind_de), diff(solar_de), fwd1(exim_op1_de), fwd1(exim_op2_de), fwd1(exim_p_de)), 2, 1, 2, 1, 7))
  #list(fwd1(vol_de), list(fwd1(wind_de), diff(solar_de), diff(gas_de))))

daily_res <- lapply(daily_forms, function(x) do.call(fitModel, x))

## Warning in arima(y, order = c(p, 0, q), seasonal = list(order = c(P, 0, :
## possible convergence problem: optim gave code = 1

## Warning in arima(y, order = c(p, 0, q), seasonal = list(order = c(P, 0, :
## possible convergence problem: optim gave code = 1

## Warning in arima(y, order = c(p, 0, q), seasonal = list(order = c(P, 0, :
## possible convergence problem: optim gave code = 1

daily_res %>% 
  modelsummary

	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)	(13)	(14)	(15)	(16)	(17)	(18)
ar1	1.224	1.221	1.255	1.245	1.244	1.233	1.228	1.269	1.231	1.229	1.151	1.166	1.161	1.155	1.152	1.166	1.153	1.151
	(0.039)	(0.039)	(0.038)	(0.039)	(0.039)	(0.053)	(0.054)	(0.051)	(0.054)	(0.053)	(0.036)	(0.036)	(0.031)	(0.048)	(0.024)	(0.036)	(0.036)	(0.038)
ar2	−0.253	−0.250	−0.279	−0.273	−0.272	−0.268	−0.264	−0.297	−0.268	−0.266	−0.162	−0.175	−0.171	−0.166	−0.163	−0.176	−0.164	−0.163
	(0.033)	(0.032)	(0.033)	(0.033)	(0.033)	(0.041)	(0.042)	(0.041)	(0.042)	(0.042)	(0.034)	(0.034)	(0.032)	(0.040)	(0.025)	(0.034)	(0.035)	(0.035)
ma1	−0.864	−0.863	−0.870	−0.867	−0.867	−0.838	−0.836	−0.852	−0.833	−0.833	−0.916	−0.917	−0.919	−0.915	−0.917	−0.915	−0.914	−0.928
	(0.029)	(0.029)	(0.026)	(0.028)	(0.028)	(0.043)	(0.044)	(0.040)	(0.044)	(0.043)	(0.018)	(0.019)	(0.015)	(0.031)	(0.015)	(0.018)	(0.019)	(0.021)
sar1	1.071	1.071	1.081	1.075	1.075	1.105	1.106	1.107	1.105	1.106	1.177	1.172	1.175	1.174	1.177	1.172	1.174	1.169
	(0.025)	(0.025)	(0.025)	(0.026)	(0.026)	(0.025)	(0.025)	(0.025)	(0.025)	(0.025)	(0.032)	(0.032)	(0.030)	(0.034)	(0.032)	(0.032)	(0.032)	(0.032)
sar2	−0.073	−0.072	−0.082	−0.077	−0.077	−0.106	−0.107	−0.107	−0.105	−0.106	−0.177	−0.172	−0.175	−0.174	−0.177	−0.172	−0.174	−0.169
	(0.025)	(0.025)	(0.025)	(0.025)	(0.025)	(0.025)	(0.025)	(0.025)	(0.025)	(0.025)	(0.032)	(0.032)	(0.030)	(0.034)	(0.032)	(0.032)	(0.032)	(0.032)
sma1	−0.980	−0.980	−0.983	−0.979	−0.979	−0.989	−0.990	−0.990	−0.988	−0.989	−0.990	−0.991	−0.987	−0.989	−0.989	−0.991	−0.990	−0.989
	(0.008)	(0.008)	(0.007)	(0.008)	(0.008)	(0.006)	(0.006)	(0.006)	(0.006)	(0.006)	(0.005)	(0.000)	(0.010)	(0.019)	(0.024)	(0.009)	(0.001)	(0.001)
intercept	2.392	1.708	2.253	2.357	1.661	2.207	1.714	2.101	2.155	1.729	1.916	2.270	2.001	2.085	2.322	2.243	2.277	1.931
	(0.344)	(0.318)	(0.402)	(0.343)	(0.309)	(0.440)	(0.444)	(0.489)	(0.418)	(0.421)	(0.551)	(0.605)	(0.568)	(0.601)	(0.531)	(0.570)	(0.572)	(0.467)
ext	−0.089	−0.087				−0.070	−0.064				0.033	−0.038		0.017	0.035	−0.047	0.020
	(0.016)	(0.016)				(0.015)	(0.015)				(0.015)	(0.019)		(0.023)	(0.015)	(0.019)	(0.022)
wind_dk1			−0.073	−0.091
			(0.019)	(0.019)
exim_op1_dk1			0.064	0.100	0.102
			(0.033)	(0.033)	(0.033)
exim_op2_dk1			−0.224
			(0.031)
exim_p_dk1			0.078	−0.081	−0.085
			(0.037)	(0.031)	(0.031)
wind_pen_dk1					−0.088
					(0.020)
wind_dk2								−0.052	−0.060
								(0.016)	(0.017)
exim_op1_dk2								−0.015	−0.007	−0.012
								(0.042)	(0.043)	(0.043)
exim_op2_dk2								−0.304
								(0.040)
exim_p_dk2								0.145	−0.047	−0.052
								(0.048)	(0.042)	(0.042)
wind_pen_dk2										−0.054
										(0.017)
ext1													0.030					0.037
													(0.015)					(0.015)
ext2													−0.035					−0.020
													(0.019)					(0.019)
ext3																		0.085
																		(0.012)
ext4																		0.022
																		(0.013)
ext5																		−0.078
																		(0.014)
Num.Obs.	1826	1826	1826	1826	1826	1826	1826	1826	1826	1826	1096	1095	1095	1096	1096	1095	1096	1095
AIC	2880.8	2883.5	2822.5	2873.6	2875.7	3161.9	3165.3	3108.3	3164.2	3166.8	489.7	490.6	488.6	494.2	489.1	488.4	493.9	436.1
BIC	2930.4	2933.1	2888.6	2934.2	2936.3	3211.5	3214.9	3174.4	3224.8	3227.4	534.7	535.6	538.6	539.2	534.1	533.4	538.9	501.1
RMSE	0.53	0.53	0.52	0.53	0.53	0.57	0.57	0.56	0.57	0.57	0.30	0.30	0.30	0.30	0.30	0.30	0.30	0.29

Hourly price volatility model with gets

This is where we have our current comparative advantage in that we can model hourly price volatility directly.

Data

# Loads hourly price data for de, dk1, dk2
for(cty in c('dk1', 'dk2', 'de')){
  temp <- read_xlsx(here('code/rintamaki_replication/data/raw_data.xlsx'), sheet = cty) %>% 
    clean_names
  assign(paste0("data_", cty), temp)
}

Germany

data_de %>% 
  ggplot(aes(x = date, y = de_price)) +
  geom_line(linewidth = 0.1) +
  labs(title = "Germany: Hourly Electricity Prices", y = "Hourly Electricity Price", x = "Date")

# Check to make sure data is ordered
data_de %>% arrange(date, hour) %>% identical(data_de)

## [1] TRUE

# The below model incorporates ARCH(24) to be selected over, available regressors for wind forecasts, solar forecasts, and imports to and from France. In line with Rintamaki et al, we incorporate moving average terms for half-day, daily, and weekly prices volatility (levels?). Seasonal moving average terms are not yet incorporated here as they are in Rintamaki et al. Rather we might want to consider the seasonal moving average to have a periodicity of 24 hours (moving average over the same time of day)
de_mod <- arx(data_de$de_price, 
              # Include 24 ARCH terms to be selected over
              # Question: In theory, the ARCH(24) would proxy the seasonal (daily) AR term?
              arch = 1:24, 
              # Additional regressors to select over (only wind forecasts, solar forecasts, and imports between Germany and France are available at hourly frequency in the replication data)
              # Worth doing additional data scoping
              # Also perhaps worth including additional indicators for time of day (assuming this variation is not mopped up by the moving average terms)
              vxreg = as.matrix(data_de[c('de_wind_fcast', 'de_solar_fcast', 'de_fr', 'fr_de')]), 
              # As in Rintamaki et al. the below line incorporates moving average terms for the half-day (12), day (24 hours) and week (168 hours) 
              log.ewma = c(12, 24, 168)) 

## Warning in arx(data_de$de_price, arch = 1:24, vxreg = as.matrix(data_de[c("de_wind_fcast", : 
## 
## New default 'mc = TRUE' in arx() as of version 0.28
## This warning only appears the first time arx() is invoked
## To suppress this warning, set options(mc.warning = FALSE)

# Perform gets model selection over the log-variance model above
# Turn of ARCH diagnostics as the GUM does not pass (could possible manipulate lag and p value choice here)
de_vmod <- getsv(de_mod, t.pval = 0.001, ar.LjungB = NULL)

## GUM log-variance equation:
## 
##                reg.no. keep        coef   std.error   t-stat   p-value    
## vconst               1    1  1.8954e+00  1.0014e-01 358.2185 < 2.2e-16 ***
## arch1                2    0  5.8181e-01  6.2374e-03  93.2783 < 2.2e-16 ***
## arch2                3    0 -3.5317e-03  7.2259e-03  -0.4888 0.6250156    
## arch3                4    0 -3.5501e-02  7.2242e-03  -4.9142 8.967e-07 ***
## arch4                5    0 -1.0043e-02  7.2186e-03  -1.3913 0.1641497    
## arch5                6    0 -3.3139e-02  7.2133e-03  -4.5942 4.364e-06 ***
## arch6                7    0 -1.1185e-02  7.2128e-03  -1.5506 0.1210008    
## arch7                8    0 -3.2399e-02  7.2134e-03  -4.4915 7.103e-06 ***
## arch8                9    0 -8.7058e-03  7.2241e-03  -1.2051 0.2281732    
## arch9               10    0 -1.3099e-03  7.2319e-03  -0.1811 0.8562671    
## arch10              11    0 -1.6037e-02  7.2305e-03  -2.2180 0.0265629 *  
## arch11              12    0 -1.0092e-03  7.2273e-03  -0.1396 0.8889439    
## arch12              13    0 -2.0967e-02  7.1982e-03  -2.9128 0.0035855 ** 
## arch13              14    0 -1.8687e-02  7.1691e-03  -2.6067 0.0091480 ** 
## arch14              15    0 -7.0670e-03  7.1822e-03  -0.9840 0.3251394    
## arch15              16    0 -2.2407e-02  7.1833e-03  -3.1193 0.0018146 ** 
## arch16              17    0 -3.7778e-03  7.1829e-03  -0.5259 0.5989277    
## arch17              18    0 -2.3916e-02  7.1811e-03  -3.3304 0.0008684 ***
## arch18              19    0  8.3342e-03  7.1800e-03   1.1608 0.2457487    
## arch19              20    0 -4.5284e-03  7.1834e-03  -0.6304 0.5284426    
## arch20              21    0 -5.3013e-03  7.1845e-03  -0.7379 0.4605924    
## arch21              22    0 -9.2692e-04  7.1901e-03  -0.1289 0.8974256    
## arch22              23    0  1.3481e-02  7.1919e-03   1.8744 0.0608876 .  
## arch23              24    0  6.4164e-02  7.1902e-03   8.9239 < 2.2e-16 ***
## arch24              25    0  1.2186e-01  6.1825e-03  19.7100 < 2.2e-16 ***
## logEqWMA(12)        26    0  3.0409e-01  2.7297e-02  11.1402 < 2.2e-16 ***
## logEqWMA(24)        27    0  4.1168e-02  3.4000e-02   1.2108 0.2259724    
## logEqWMA(168)       28    0 -3.9480e-03  2.1543e-02  -0.1833 0.8545958    
## de_wind_fcast       29    0  7.4541e-06  2.2588e-06   3.3000 0.0009681 ***
## de_solar_fcast      30    0 -4.8653e-06  1.9912e-06  -2.4434 0.0145561 *  
## de_fr               31    0 -1.5198e-04  1.1846e-05 -12.8305 < 2.2e-16 ***
## fr_de               32    0 -5.7088e-05  2.0817e-05  -2.7424 0.0061024 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Diagnostics:
## 
##                       Chi-sq df p-value
## Ljung-Box AR(1)    3325.5273  1       0
## Ljung-Box ARCH(25)    3.8785 25       1

## 21 path(s) to search

## Searching: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

## 
##   Path 1: 3 22 12 10 28 17 20 15 19 21 27 5 9 7 23 30 32 13 
##   Path 2: 5 12 22 10 28 3 17 20 15 19 21 27 9 7 23 30 32 13 
##   Path 3: 7 12 22 10 28 3 17 20 15 19 21 27 5 9 23 30 32 13 
##   Path 4: 9 12 22 28 3 17 20 10 15 19 21 27 5 7 23 30 32 13 
##   Path 5: 10 22 12 28 3 17 20 15 19 21 27 5 9 7 23 30 32 13 
##   Path 6: 11 28 22 3 17 20 15 19 21 27 12 9 5 7 10 23 32 30 -14 
##   Path 7: 12 22 28 10 3 17 20 15 19 21 27 5 9 7 23 30 32 13 
##   Path 8: 13 10 28 22 3 17 20 19 15 21 27 5 9 12 7 23 30 32 
##   Path 9: 14 22 28 12 10 17 3 20 27 21 19 9 5 7 15 23 30 32 -11 
##   Path 10: 15 22 12 28 10 3 17 20 19 21 27 5 9 7 23 30 32 13 
##   Path 11: 16 28 22 12 10 3 20 27 21 19 9 5 7 17 23 14 -30 -32 -11 
##   Path 12: 17 22 12 28 10 3 20 15 19 21 27 5 9 7 23 30 32 13 
##   Path 13: 19 20 12 22 10 28 3 17 15 21 27 5 9 7 23 30 32 13 
##   Path 14: 20 22 12 28 10 3 17 15 19 21 27 5 9 7 23 30 32 13 
##   Path 15: 21 12 28 10 3 22 17 15 27 20 19 5 9 7 23 30 32 13 
##   Path 16: 22 12 28 10 3 17 20 15 19 21 27 5 9 7 23 30 32 13 
##   Path 17: 23 12 10 28 3 17 20 22 21 19 15 5 9 7 27 30 32 13 
##   Path 18: 27 22 12 10 28 17 20 3 15 21 19 5 9 7 23 30 32 13 
##   Path 19: 28 22 12 10 3 17 20 15 19 21 27 5 9 7 23 30 32 13 
##   Path 20: 30 22 12 28 10 17 3 20 15 19 21 27 5 9 7 23 32 13 
##   Path 21: 32 28 22 12 10 3 17 20 15 27 21 19 5 9 7 23 30 13 
## 
## Terminal models:
## 
##         info(sc)      logl     n  k
## spec 1: 8.386333 -109534.0 26139 14
## spec 2: 8.394811 -109644.8 26139 14
## spec 3: 8.385634 -109524.8 26139 14
## spec 4: 8.380971 -109453.7 26139 16
## 
## Retained regressors (final model):
## 
##   vconst   arch1   arch3   arch5   arch7   arch10   arch12   arch14   arch17   arch23   arch24   logEqWMA(12)   de_wind_fcast   de_solar_fcast   de_fr   fr_de

de_vmod %>% plot

DK1

data_dk1 %>% 
  ggplot(aes(x = date, y = dk1_price)) +
  geom_line(linewidth = 0.1) +
  labs(main = "DK1: Hourly Electricity Prices")

data_dk1 %>% arrange(date, hour) %>% identical(data_dk1)

## [1] TRUE

# Same spec as above with appropriate corrections to the regressors
dk1_mod <- arx(data_dk1$dk1_price, 
              arch = 1:24, 
              vxreg = as.matrix(data_dk1[c("dk1_load_fcast", "dk1_wind_fcast", "dk1_no2", "no2_dk1", "dk1_se3", "se3_dk1")]),
               log.ewma = c(12, 24, 168)) 


# Perform gets model selection over the log-variance model above
dk1_vmod <- getsv(dk1_mod, t.pval = 0.001, ar.LjungB = NULL)

## GUM log-variance equation:
## 
##                reg.no. keep        coef   std.error    t-stat   p-value    
## vconst               1    1  2.0172e+00  5.4321e-02 1379.0242 < 2.2e-16 ***
## arch1                2    0  6.0445e-01  4.8399e-03  124.8876 < 2.2e-16 ***
## arch2                3    0  3.0277e-02  5.6434e-03    5.3650 8.136e-08 ***
## arch3                4    0 -1.7644e-02  5.6403e-03   -3.1282  0.001760 ** 
## arch4                5    0 -1.2948e-02  5.6355e-03   -2.2977  0.021586 *  
## arch5                6    0 -1.7261e-02  5.6325e-03   -3.0646  0.002181 ** 
## arch6                7    0  1.4257e-03  5.6321e-03    0.2531  0.800157    
## arch7                8    0 -4.6656e-03  5.6308e-03   -0.8286  0.407345    
## arch8                9    0 -1.1409e-02  5.6288e-03   -2.0268  0.042686 *  
## arch9               10    0  3.1481e-03  5.6334e-03    0.5588  0.576277    
## arch10              11    0 -7.0138e-03  5.6371e-03   -1.2442  0.213423    
## arch11              12    0 -1.3609e-02  5.6385e-03   -2.4137  0.015797 *  
## arch12              13    0 -2.3295e-02  5.6207e-03   -4.1445 3.411e-05 ***
## arch13              14    0  1.6176e-02  5.5976e-03    2.8898  0.003857 ** 
## arch14              15    0 -9.1378e-03  5.6031e-03   -1.6309  0.102929    
## arch15              16    0  3.6041e-03  5.6050e-03    0.6430  0.520217    
## arch16              17    0 -2.7019e-03  5.6056e-03   -0.4820  0.629813    
## arch17              18    0  9.9656e-03  5.6073e-03    1.7773  0.075533 .  
## arch18              19    0 -6.3015e-03  5.6078e-03   -1.1237  0.261142    
## arch19              20    0  6.9875e-03  5.6080e-03    1.2460  0.212770    
## arch20              21    0  1.1826e-02  5.6071e-03    2.1092  0.034936 *  
## arch21              22    0  1.8614e-02  5.6071e-03    3.3197  0.000902 ***
## arch22              23    0  2.4458e-02  5.6079e-03    4.3613 1.296e-05 ***
## arch23              24    0  7.6882e-02  5.6049e-03   13.7168 < 2.2e-16 ***
## arch24              25    0  8.5187e-02  4.7824e-03   17.8126 < 2.2e-16 ***
## logEqWMA(12)        26    0  1.9586e-01  1.7495e-02   11.1952 < 2.2e-16 ***
## logEqWMA(24)        27    0 -4.7579e-02  1.8586e-02   -2.5600  0.010470 *  
## logEqWMA(168)       28    0  2.2026e-02  1.0175e-02    2.1647  0.030418 *  
## dk1_load_fcast      29    0  2.9724e-07  1.4970e-05    0.0199  0.984158    
## dk1_wind_fcast      30    0  7.0917e-05  1.0943e-05    6.4808 9.219e-11 ***
## dk1_no2             31    0 -6.9258e-05  2.3247e-05   -2.9792  0.002892 ** 
## no2_dk1             32    0 -1.3200e-04  2.3125e-05   -5.7080 1.150e-08 ***
## dk1_se3             33    0  5.4250e-05  3.6553e-05    1.4842  0.137774    
## se3_dk1             34    0 -2.9485e-06  3.3375e-05   -0.0883  0.929602    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Diagnostics:
## 
##                       Chi-sq df p-value
## Ljung-Box AR(1)    2396.4839  1 0.00000
## Ljung-Box ARCH(25)    5.7287 25 0.99998

## 23 path(s) to search

## Searching: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

## 
##   Path 1: 4 29 34 7 17 16 10 8 11 19 20 33 15 18 28 27 14 -31 -21 -9 -12 -22 
##   Path 2: 5 29 34 7 17 16 10 8 11 19 20 33 15 18 28 27 14 31 -9 -21 -12 -22 
##   Path 3: 6 29 34 17 16 10 8 11 19 20 33 15 18 7 28 27 14 -31 -12 -21 -4 -22 
##   Path 4: 7 29 34 17 16 10 8 11 19 20 33 15 18 28 27 5 14 31 -9 -21 -12 -22 
##   Path 5: 8 29 34 7 17 16 10 11 19 20 33 15 18 28 27 5 14 31 -9 -21 -12 -22 
##   Path 6: 9 29 34 7 10 17 16 19 20 11 33 15 18 5 28 27 8 14 -31 -21 -22 
##   Path 7: 10 29 34 7 17 16 8 11 19 20 33 15 18 28 27 5 14 31 -9 -21 -12 -22 
##   Path 8: 11 29 10 34 7 17 16 8 19 20 33 15 18 28 27 5 14 31 -9 -21 -12 -22 
##   Path 9: 12 29 34 7 17 16 10 8 19 20 33 15 18 5 28 27 14 11 -31 -21 -22 
##   Path 10: 14 29 34 7 15 17 16 10 8 11 19 20 33 18 27 28 5 31 -9 -21 -12 -22 
##   Path 11: 15 29 16 34 7 10 17 8 11 19 20 33 18 28 27 5 14 31 -9 -21 -12 -22 
##   Path 12: 16 29 34 17 7 10 8 11 19 20 33 15 18 28 27 5 14 31 -9 -21 -12 -22 
##   Path 13: 17 29 34 7 16 10 8 11 19 20 33 15 18 28 27 5 14 31 -9 -21 -12 -22 
##   Path 14: 18 29 34 7 17 19 10 8 16 11 15 20 33 28 27 5 14 31 -9 -21 -12 -22 
##   Path 15: 19 29 34 7 17 16 10 8 20 11 33 15 18 28 27 5 14 31 -9 -21 -12 -22 
##   Path 16: 20 29 34 7 17 16 10 19 8 11 33 15 18 28 27 5 14 31 -9 -21 -12 -22 
##   Path 17: 21 29 34 7 17 16 10 8 19 11 18 15 33 28 27 14 5 20 -31 -12 -9 
##   Path 18: 27 29 34 7 16 17 10 8 11 20 19 18 33 28 15 5 14 31 -9 -21 -12 -22 
##   Path 19: 28 34 29 7 17 16 10 8 11 19 20 15 18 33 27 5 14 31 -9 -21 -12 -22 
##   Path 20: 29 34 7 17 16 10 8 11 19 20 33 15 18 28 27 5 14 31 -9 -21 -12 -22 
##   Path 21: 31 34 7 29 17 16 10 8 11 19 20 33 15 18 28 27 5 14 -9 -21 -12 -22 
##   Path 22: 33 29 7 17 16 10 34 8 11 19 20 15 18 28 27 5 14 31 -9 -21 -12 -22 
##   Path 23: 34 29 7 17 16 10 8 11 19 20 33 15 18 28 27 5 14 31 -9 -21 -12 -22 
## 
## Terminal models:
## 
##         info(sc)      logl     n  k
## spec 1: 8.671951 -189222.2 43661 17
## spec 2: 8.687761 -189572.7 43661 16
## spec 3: 8.687266 -189556.5 43661 17
## spec 4: 8.692836 -189683.5 43661 16
## spec 5: 8.686429 -189543.6 43661 16
## spec 6: 8.682763 -189463.6 43661 16
## 
## Retained regressors (final model):
## 
##   vconst   arch1   arch2   arch4   arch5   arch8   arch11   arch12   arch20   arch21   arch22   arch23   arch24   logEqWMA(12)   dk1_wind_fcast   dk1_no2   no2_dk1

dk1_vmod %>% plot

DK2

data_dk2 %>% 
  ggplot(aes(x = date, y = dk2_price)) +
  geom_line(linewidth = 0.1) +
  labs(main = "DK2: Hourly Electricity Prices")

data_dk2 %>% arrange(date, hour) %>% identical(data_dk2)

## [1] TRUE

# Same spec as above with appropriate corrections to the regressors
dk2_mod <- arx(data_dk2$dk2_price, 
              arch = 1:24, 
              vxreg = as.matrix(data_dk2[c("dk2_load_fcast", "dk2_wind_fcast", 
                                 "dk2_se4","dk2_se4_2")]),
               log.ewma = c(12, 24, 168)) 


# Perform gets model selection over the log-variance model above
dk2_vmod <- getsv(dk2_mod, t.pval = 0.001, ar.LjungB = NULL)

## GUM log-variance equation:
## 
##                reg.no. keep        coef   std.error    t-stat   p-value    
## vconst               1    1  2.2187e+00  4.4496e-02 2486.3454 < 2.2e-16 ***
## arch1                2    0  6.2910e-01  4.8220e-03  130.4648 < 2.2e-16 ***
## arch2                3    0  3.0162e-04  5.6846e-03    0.0531  0.957685    
## arch3                4    0 -1.2263e-02  5.6794e-03   -2.1592  0.030839 *  
## arch4                5    0 -1.2063e-02  5.6739e-03   -2.1260  0.033507 *  
## arch5                6    0 -5.9613e-03  5.6698e-03   -1.0514  0.293074    
## arch6                7    0 -1.5173e-02  5.6686e-03   -2.6767  0.007438 ** 
## arch7                8    0  1.8626e-03  5.6688e-03    0.3286  0.742487    
## arch8                9    0 -1.9672e-03  5.6704e-03   -0.3469  0.728651    
## arch9               10    0 -4.3919e-03  5.6709e-03   -0.7745  0.438667    
## arch10              11    0  1.0484e-02  5.6759e-03    1.8472  0.064731 .  
## arch11              12    0 -2.7936e-02  5.6785e-03   -4.9196 8.702e-07 ***
## arch12              13    0 -1.6450e-02  5.6647e-03   -2.9040  0.003686 ** 
## arch13              14    0  9.6957e-03  5.6439e-03    1.7179  0.085822 .  
## arch14              15    0  2.6291e-03  5.6485e-03    0.4654  0.641614    
## arch15              16    0  4.4085e-04  5.6506e-03    0.0780  0.937815    
## arch16              17    0 -1.3189e-03  5.6509e-03   -0.2334  0.815451    
## arch17              18    0  4.9549e-03  5.6517e-03    0.8767  0.380651    
## arch18              19    0 -1.5177e-03  5.6527e-03   -0.2685  0.788322    
## arch19              20    0 -3.0555e-04  5.6523e-03   -0.0541  0.956889    
## arch20              21    0  1.6525e-02  5.6521e-03    2.9237  0.003460 ** 
## arch21              22    0  1.6383e-02  5.6523e-03    2.8984  0.003753 ** 
## arch22              23    0  3.1062e-02  5.6520e-03    5.4958 3.910e-08 ***
## arch23              24    0  8.3581e-02  5.6530e-03   14.7851 < 2.2e-16 ***
## arch24              25    0  1.0642e-01  4.7691e-03   22.3151 < 2.2e-16 ***
## logEqWMA(12)        26    0  1.8053e-01  1.6245e-02   11.1126 < 2.2e-16 ***
## logEqWMA(24)        27    0 -7.4563e-02  1.7611e-02   -4.2339 2.302e-05 ***
## logEqWMA(168)       28    0  2.2178e-02  9.5550e-03    2.3211  0.020285 *  
## dk2_load_fcast      29    0 -1.0045e-05  1.9782e-05   -0.5078  0.611615    
## dk2_wind_fcast      30    0  1.3429e-04  3.0722e-05    4.3712 1.239e-05 ***
## dk2_se4             31    0  7.6436e-06  2.1887e-05    0.3492  0.726915    
## dk2_se4_2           32    0 -1.1370e-04  1.6364e-05   -6.9482 3.752e-12 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Diagnostics:
## 
##                        Chi-sq df p-value
## Ljung-Box AR(1)    1633.88530  1       0
## Ljung-Box ARCH(25)    0.60547 25       1

## 22 path(s) to search

## Searching: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

## 
##   Path 1: 3 20 16 17 19 8 9 31 15 29 18 10 6 11 28 4 14 13 22 -27 
##   Path 2: 4 20 16 17 9 8 31 19 29 15 10 18 3 6 11 28 14 13 22 -27 
##   Path 3: 5 3 20 16 17 9 8 31 19 15 29 10 18 11 6 28 14 13 22 -27 
##   Path 4: 6 16 20 3 17 8 9 31 19 15 29 10 18 11 28 4 14 13 22 -27 
##   Path 5: 7 20 16 3 17 19 31 9 15 29 18 10 8 11 5 28 14 13 22 -27 
##   Path 6: 8 3 20 16 17 9 19 31 15 29 18 10 6 11 28 4 14 13 22 -27 
##   Path 7: 9 20 3 16 8 17 19 31 15 29 18 10 6 11 28 4 14 13 22 -27 
##   Path 8: 10 3 20 16 17 8 19 31 15 29 9 18 6 11 28 4 14 13 22 -27 
##   Path 9: 11 3 10 20 16 17 8 9 19 31 15 29 18 6 28 4 14 13 22 -27 
##   Path 10: 13 3 16 20 17 9 8 19 31 14 29 18 10 6 15 11 28 4 22 -27 
##   Path 11: 14 3 20 16 17 8 9 31 19 29 18 10 6 11 15 13 28 4 22 -27 
##   Path 12: 15 3 20 17 16 8 9 19 31 29 18 10 6 11 28 4 14 13 22 -27 
##   Path 13: 16 3 20 17 19 8 9 31 15 29 18 10 6 11 28 4 14 13 22 -27 
##   Path 14: 17 16 3 20 19 8 9 31 15 29 18 10 6 11 28 4 14 13 22 -27 
##   Path 15: 18 20 3 16 19 17 9 8 31 29 15 10 6 11 28 4 14 13 22 -27 
##   Path 16: 19 3 16 20 17 8 9 31 15 29 18 10 6 11 28 4 14 13 22 -27 
##   Path 17: 20 3 16 17 19 8 9 31 15 29 18 10 6 11 28 4 14 13 22 -27 
##   Path 18: 21 3 16 19 17 8 9 31 15 29 18 10 6 11 20 28 4 14 13 -27 
##   Path 19: 22 16 3 20 17 9 8 19 31 15 29 18 10 6 11 28 4 14 13 -27 
##   Path 20: 28 3 20 16 29 17 8 9 19 15 31 18 10 6 11 4 14 13 22 -27 
##   Path 21: 29 20 3 16 17 19 8 9 31 15 18 10 6 11 28 4 14 13 22 -27 
##   Path 22: 31 20 3 16 17 19 8 9 15 29 18 10 6 11 28 4 14 13 22 -27 
## 
## Terminal models:
## 
##         info(sc)      logl     n  k
## spec 1: 9.078499 -198118.7 43661 13
## spec 2: 9.107696 -198756.1 43661 13
## spec 3: 9.109620 -198798.1 43661 13
## spec 4: 9.077464 -198096.1 43661 13
## 
## Retained regressors (final model):
## 
##   vconst   arch1   arch4   arch6   arch11   arch21   arch22   arch23   arch24   logEqWMA(12)   logEqWMA(24)   dk2_wind_fcast   dk2_se4_2

dk2_vmod %>% plot

Replicate Weekly

Again, in the specification looking at weekly price volatility the dependent variable of interest is the standard deviation of daily prices per week. We have the advantage of being able to model the volatility of daily prices directly with gets.

Data

Pulled from replication data available at:https://doi-org.ezproxy-prd.bodleian.ox.ac.uk/10.1016/j.eneco.2016.12.019

# Required including "here("code/rintamaki_replication/data/" to data read commands in the listed files
source(here("code/rintamaki_replication/models/read_weekly_data.R"))

Models as in Rintamäki et al.

weekly_forms <- list(
  # SPECS FOR DK1
  list(vol_dk1, wind_dk1, 1, 0, 0, 1, 4),
  list(fwd1(vol_dk1), cbind(fwd1(wind_dk1), diff(exim_dk1)), 1, 0, 0, 1, 4),
  list(vol_dk1, wind_std_dk1, 1, 0, 0, 1, 4),
  # SPECS FOR DK2
  list(vol_dk2, wind_dk2, 1, 0, 0, 1, 4),
  list(vol_dk2, wind_std_dk2, 1, 0, 0, 1, 4),
  # SPECS FOR DE
  list(fwd1(vol_de), diff(wind_de), 1, 0, 0, 0, 4),
  list(fwd1(vol_de), diff(solar_de), 1, 0, 0, 0, 4),
  list(vol_de, res_de, 1, 0, 0, 0, 4),
  list(fwd1(vol_de), cbind(diff(wind_de), fwd1(exim_de)), 1, 0, 0, 0, 4))

weekly_res <- lapply(weekly_forms, function(x) do.call(fitModel, x))

weekly_res %>%
  modelsummary

	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
ar1	0.307	0.304	0.303	0.345	0.346	0.168	0.206	0.173	0.156
	(0.059)	(0.060)	(0.059)	(0.060)	(0.060)	(0.082)	(0.079)	(0.080)	(0.084)
sma1	0.199	0.203	0.188	0.171	0.150
	(0.062)	(0.063)	(0.061)	(0.068)	(0.067)
intercept	0.402	0.522	−0.593	1.317	0.760	1.967	1.968	0.764	1.941
	(0.474)	(0.526)	(0.444)	(0.390)	(0.371)	(0.040)	(0.042)	(1.071)	(0.052)
ext	0.182		0.365	0.062	0.184	0.105	−0.171	0.132
	(0.071)		(0.073)	(0.072)	(0.076)	(0.059)	(0.113)	(0.117)
ext1		0.164							0.110
		(0.079)							(0.060)
ext2		0.041							0.028
		(0.076)							(0.037)
Num.Obs.	261	260	261	261	261	155	155	156	155
AIC	429.4	430.3	412.0	500.6	495.5	173.0	174.0	175.7	174.4
BIC	447.2	451.6	429.8	518.4	513.3	185.1	186.1	187.9	189.6
RMSE	0.54	0.54	0.52	0.62	0.61	0.41	0.41	0.41	0.41

Daily price volatility model with gets

Data

for(cty in c('dk1', 'dk2', 'de')){
  temp <- read_xlsx(here('code/rintamaki_replication/data/raw_data.xlsx'), sheet = cty) %>%
    clean_names
  assign(paste0("data_", cty), temp)
}

df_list <- list("data_de" = data_de, "data_dk1" = data_dk1, "data_dk2" = data_dk2)

for(n_df in names(df_list)){
  df <- df_list[[n_df]]
  temp <- df %>%
    tibble %>%
    # Certain days in October have multiple values for hour one across all countries...not sure why.
    # Below I take the average of the duplicate observations but might be worth looking into why this is the case
    group_by(date, hour) %>%
    summarise(across(names(df)[!(names(df) %in% c("date", "hour", "month"))], ~mean(., na.rm = TRUE))) %>%
    # Daily average price
    group_by(date) %>%
    select(-c("hour")) %>%
    summarise(across(names(df)[!(names(df) %in% c("date", "hour", "month"))], ~mean(., na.rm = TRUE)))

   assign(paste0("daily_", n_df), temp)
}

## `summarise()` has grouped output by 'date'. You can override using the
## `.groups` argument.
## `summarise()` has grouped output by 'date'. You can override using the
## `.groups` argument.
## `summarise()` has grouped output by 'date'. You can override using the
## `.groups` argument.

Germany

daily_data_de %>%
  ggplot(aes(x = date, y = de_price)) +
  geom_line(linewidth = 0.1) +
  labs(main = "Germany: Daily Electricity Prices")

daily_data_de %>% arrange(date) %>% identical(daily_data_de)

## [1] TRUE

# equally-weighted moving averages are incorporated to account for multi-day moving averages 2-, 5-, 7-, 14-day moving averages

# Might want to consider re-incorporating the seasonality dimension as well
daily_de_mod <- arx(daily_data_de$de_price,
              # Include 14 ARCH terms to be selected over (two-week period)
              # in theory, the ARHC(14) would proxy the seasonal AR term?
              arch = 1:14,
              # Additional regressors to select over (only wind forecasts, solar forecasts, and imports between Germany and France are available at hourly frequency in the replication data
              # Worth doing additional data scoping
              # Also perhaps worth including additional indicators for time of day (assuming this variation is not mopped up by the moving average terms)
              vxreg = as.matrix(daily_data_de[c('de_load_fcast', 'de_wind_fcast', 'de_solar_fcast', 'de_fr', 'fr_de')]),
              # As in Rintamaki et al. the below line incorporates moving average terms for the two-day, 5-day, one-week, and two-week prices
              log.ewma = c(2, 5, 7, 14))
# Seasonal moving average terms are not yet incorporated here as they are in Rintamaki et al. Rather we might want to consider the seasonal moving average to have a periodicity of X hours (moving average over the same time of day)

# Perform gets model selection over the log-variance model above
daily_de_vmod <- getsv(daily_de_mod, t.pval = 0.001, ar.LjungB = NULL)

## GUM log-variance equation:
## 
##                reg.no. keep        coef   std.error  t-stat   p-value    
## vconst               1    1  4.0532e+00  8.8546e-01 20.9532 4.706e-06 ***
## arch1                2    0  9.7999e-02  4.5964e-02  2.1321 0.0332313 *  
## arch2                3    0 -7.6878e-02  4.5877e-02 -1.6757 0.0940835 .  
## arch3                4    0 -6.0339e-02  3.4175e-02 -1.7656 0.0777545 .  
## arch4                5    0  5.4458e-03  3.4362e-02  0.1585 0.8741043    
## arch5                6    0 -7.6844e-02  3.3903e-02 -2.2666 0.0236161 *  
## arch6                7    0 -1.7623e-02  3.4680e-02 -0.5082 0.6114385    
## arch7                8    0  1.2392e-01  3.4654e-02  3.5760 0.0003647 ***
## arch8                9    0 -3.5032e-03  3.2051e-02 -0.1093 0.9129856    
## arch9               10    0 -7.5057e-02  3.2313e-02 -2.3228 0.0203772 *  
## arch10              11    0 -1.8537e-02  3.2130e-02 -0.5769 0.5641108    
## arch11              12    0  1.0148e-02  3.2372e-02  0.3135 0.7539700    
## arch12              13    0  1.9929e-02  3.2315e-02  0.6167 0.5375455    
## arch13              14    0  5.1869e-02  3.2050e-02  1.6184 0.1058847    
## arch14              15    0  9.6728e-02  3.1738e-02  3.0477 0.0023631 ** 
## logEqWMA(2)         16    0  1.7453e-01  1.0586e-01  1.6487 0.0994972 .  
## logEqWMA(5)         17    0 -4.6089e-01  2.0604e-01 -2.2369 0.0254993 *  
## logEqWMA(7)         18    0  9.8277e-01  2.7409e-01  3.5855 0.0003518 ***
## logEqWMA(14)        19    0 -3.2639e-01  2.2565e-01 -1.4465 0.1483359    
## de_load_fcast       20    0 -1.7656e-06  1.2515e-05 -0.1411 0.8878323    
## de_wind_fcast       21    0  1.3089e-06  1.6235e-05  0.0806 0.9357549    
## de_solar_fcast      22    0 -1.8519e-04  3.7614e-05 -4.9234 9.863e-07 ***
## de_fr               23    0 -3.5686e-04  1.3266e-04 -2.6900 0.0072587 ** 
## fr_de               24    0 -2.2958e-05  2.4425e-04 -0.0940 0.9251325    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Diagnostics:
## 
##                      Chi-sq df p-value
## Ljung-Box AR(1)    229.0275  1 0.00000
## Ljung-Box ARCH(15)   2.4452 15 0.99989

## 20 path(s) to search

## Searching: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

## 
##   Path 1: 2 24 20 21 5 9 12 7 13 11 19 14 6 4 15 
##   Path 2: 3 21 20 24 9 5 12 7 13 16 11 19 14 4 6 10 
##   Path 3: 4 20 9 24 21 5 12 7 13 11 19 14 6 3 16 10 
##   Path 4: 5 21 9 24 20 12 7 11 13 19 14 3 16 4 6 10 
##   Path 5: 6 9 20 24 21 5 13 12 11 7 14 19 4 3 16 10 
##   Path 6: 7 24 9 21 5 20 12 11 13 19 14 3 16 4 6 10 
##   Path 7: 9 21 24 20 5 12 7 11 13 19 14 3 16 4 6 10 
##   Path 8: 10 24 21 20 5 9 7 12 11 13 16 3 14 6 4 19 
##   Path 9: 11 9 24 21 5 20 12 7 13 19 14 3 16 4 6 10 
##   Path 10: 12 21 24 20 9 5 7 11 13 19 14 3 16 4 6 10 
##   Path 11: 13 21 24 5 9 20 12 7 11 19 14 3 16 4 6 10 
##   Path 12: 14 21 20 24 5 12 7 9 11 13 19 3 16 4 6 10 
##   Path 13: 15 21 20 12 24 9 5 13 7 19 11 3 16 14 4 6 
##   Path 14: 16 21 9 5 24 20 12 13 7 11 3 19 14 4 6 10 
##   Path 15: 17 9 24 21 12 7 13 5 11 20 16 3 19 14 6 18 4 
##   Path 16: 19 21 24 20 12 13 5 7 9 11 14 3 16 4 6 10 
##   Path 17: 20 21 24 9 5 12 7 11 13 19 14 3 16 4 6 10 
##   Path 18: 21 9 24 20 5 12 7 11 13 19 14 3 16 4 6 10 
##   Path 19: 23 5 9 12 20 11 7 13 21 19 14 3 16 4 6 24 
##   Path 20: 24 9 21 20 5 12 7 11 13 19 14 3 16 4 6 10 
## 
## Terminal models:
## 
##         info(sc)      logl    n k
## spec 1: 7.651125 -4107.819 1082 9
## spec 2: 7.620244 -4094.606 1082 8
## spec 3: 7.625762 -4097.591 1082 8
## spec 4: 7.696568 -4139.390 1082 7
## spec 5: 7.621981 -4095.545 1082 8
## 
## Retained regressors (final model):
## 
##   vconst   arch1   arch7   arch14   logEqWMA(5)   logEqWMA(7)   de_solar_fcast   de_fr

daily_de_vmod %>% plot

DK1

daily_data_dk1 %>%
  ggplot(aes(x = date, y = dk1_price)) +
  geom_line(linewidth = 0.1) +
  labs(main = "DK1: Daily Electricity Prices")

daily_data_dk1 %>% arrange(date) %>% identical(daily_data_dk1)

## [1] TRUE

# Same spec as above with appropriate corrections to the regressors
daily_dk1_mod <- arx(daily_data_dk1$dk1_price,
              mc = TRUE,
              arch = 1:14,
              vxreg = as.matrix(daily_data_dk1[c("dk1_load_fcast", "dk1_wind_fcast", "dk1_no2", "no2_dk1", "dk1_se3", "se3_dk1")]),
               log.ewma = c(2, 5, 7, 14))


# Perform gets model selection over the log-variance model above
daily_dk1_vmod <- getsv(daily_dk1_mod, t.pval = 0.001, ar.LjungB = NULL)

## GUM log-variance equation:
## 
##                reg.no. keep        coef   std.error  t-stat   p-value    
## vconst               1    1  2.9178e+00  4.1339e-01 49.8190 1.686e-12 ***
## arch1                2    0  1.9989e-01  3.6222e-02  5.5186 3.916e-08 ***
## arch2                3    0 -7.8018e-03  3.7406e-02 -0.2086  0.834806    
## arch3                4    0 -7.7115e-03  2.6489e-02 -0.2911  0.770988    
## arch4                5    0  3.6879e-02  2.7116e-02  1.3600  0.173986    
## arch5                6    0  4.9349e-02  2.6936e-02  1.8321  0.067102 .  
## arch6                7    0  6.8063e-02  2.6755e-02  2.5440  0.011044 *  
## arch7                8    0  1.0541e-01  2.7226e-02  3.8717  0.000112 ***
## arch8                9    0  8.1658e-03  2.4635e-02  0.3315  0.740326    
## arch9               10    0  2.0706e-02  2.4692e-02  0.8386  0.401808    
## arch10              11    0  9.5798e-03  2.4505e-02  0.3909  0.695892    
## arch11              12    0 -4.8314e-02  2.4531e-02 -1.9695  0.049047 *  
## arch12              13    0  3.2894e-02  2.4638e-02  1.3351  0.182019    
## arch13              14    0  2.2363e-02  2.4645e-02  0.9074  0.364330    
## arch14              15    0  6.1874e-02  2.3742e-02  2.6061  0.009234 ** 
## logEqWMA(2)         16    0  2.2986e-01  8.5741e-02  2.6809  0.007411 ** 
## logEqWMA(5)         17    0 -6.7038e-02  1.4189e-01 -0.4725  0.636658    
## logEqWMA(7)         18    0  7.5677e-02  1.5598e-01  0.4852  0.627613    
## logEqWMA(14)        19    0 -5.8125e-02  1.0396e-01 -0.5591  0.576172    
## dk1_load_fcast      20    0 -2.5975e-04  1.4978e-04 -1.7342  0.083048 .  
## dk1_wind_fcast      21    0  2.1748e-04  7.5974e-05  2.8626  0.004251 ** 
## dk1_no2             22    0 -1.9722e-04  2.1216e-04 -0.9296  0.352704    
## no2_dk1             23    0 -4.6579e-04  2.0719e-04 -2.2482  0.024687 *  
## dk1_se3             24    0  6.7955e-04  2.9997e-04  2.2654  0.023607 *  
## se3_dk1             25    0 -3.5859e-04  2.7468e-04 -1.3055  0.191907    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Diagnostics:
## 
##                      Chi-sq df p-value
## Ljung-Box AR(1)    338.7984  1 0.00000
## Ljung-Box ARCH(15)   4.9475 15 0.99256

## 22 path(s) to search

## Searching: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

## 
##   Path 1: 3 4 9 11 19 18 17 14 10 22 25 5 13 12 20 6 24 
##   Path 2: 4 3 9 11 19 18 17 14 10 22 25 5 13 12 20 6 24 
##   Path 3: 5 17 3 4 9 11 18 19 14 22 10 25 13 12 20 6 24 
##   Path 4: 6 17 3 9 11 4 19 18 14 22 10 25 20 13 12 5 24 
##   Path 5: 7 3 4 11 9 19 22 10 14 25 17 18 5 20 12 13 24 6 
##   Path 6: 9 3 4 11 19 18 17 14 10 22 25 5 13 12 20 6 24 
##   Path 7: 10 3 4 19 18 9 11 17 14 22 25 5 13 12 20 6 24 
##   Path 8: 11 3 4 9 19 18 17 14 10 22 25 5 13 12 20 6 24 
##   Path 9: 12 11 3 4 9 17 18 10 19 14 22 13 5 25 20 6 24 
##   Path 10: 13 3 4 19 18 9 11 17 10 22 14 25 5 12 20 6 24 
##   Path 11: 14 3 4 11 19 9 18 17 10 22 25 5 13 12 20 6 24 
##   Path 12: 15 19 3 18 11 4 9 17 22 10 5 25 13 12 20 14 6 24 
##   Path 13: 16 11 18 19 9 4 5 14 22 10 25 17 20 13 12 6 24 
##   Path 14: 17 3 18 9 11 4 19 14 10 22 25 5 13 12 20 6 24 
##   Path 15: 18 17 3 9 11 4 19 14 10 22 25 5 13 12 20 6 24 
##   Path 16: 19 3 18 9 4 11 17 14 10 22 25 5 13 12 20 6 24 
##   Path 17: 20 3 9 4 11 19 18 17 14 10 22 25 5 13 12 6 24 
##   Path 18: 21 4 11 9 19 18 3 17 14 22 10 5 25 20 13 12 6 
##   Path 19: 22 3 9 4 11 19 18 17 14 10 25 5 13 12 20 6 24 
##   Path 20: 23 3 4 9 11 18 19 17 22 14 10 5 13 12 25 20 6 24 
##   Path 21: 24 3 9 11 4 19 18 17 22 10 14 5 13 12 20 25 6 
##   Path 22: 25 3 9 4 11 19 18 17 14 22 10 5 13 12 20 6 24 
## 
## Terminal models:
## 
##         info(sc)      logl    n k
## spec 1: 7.938157 -7161.962 1812 8
## spec 2: 7.935530 -7163.332 1812 7
## spec 3: 7.939540 -7166.966 1812 7
## spec 4: 8.112107 -7319.561 1812 8
## spec 5: 7.907051 -7133.780 1812 8
## spec 6: 7.927551 -7156.104 1812 7
## 
## Retained regressors (final model):
## 
##   vconst   arch1   arch6   arch7   arch14   logEqWMA(2)   no2_dk1   dk1_se3

daily_dk1_vmod %>% plot

DK2

daily_data_dk2 %>%
  ggplot(aes(x = date, y = dk2_price)) +
  geom_line(linewidth = 0.1) +
  labs(main = "DK2: Daily Electricity Prices")

data_dk2 %>% arrange(date) %>% identical(data_dk2)

## [1] TRUE

# Same spec as above with appropriate corrections to the regressors
daily_dk2_mod <- arx(daily_data_dk2$dk2_price,
              arch = 1:14,
              vxreg = as.matrix(daily_data_dk2[c("dk2_load_fcast", "dk2_wind_fcast",
                                 "dk2_se4","dk2_se4_2")]),
               log.ewma = c(2, 5, 7, 14))


# Perform gets model selection over the log-variance model above
# Requires turning off both ARCH and AR Ljung-Box test - could play around with lag and pvalues
daily_dk2_vmod <- getsv(daily_dk2_mod, t.pval = 0.001, ar.LjungB = NULL, arch.LjungB = NULL)

## GUM log-variance equation:
## 
##                reg.no. keep        coef   std.error  t-stat   p-value    
## vconst               1    1  2.2735e+00  3.8929e-01 34.1078 5.214e-09 ***
## arch1                2    0  1.3671e-01  3.3986e-02  4.0226 5.995e-05 ***
## arch2                3    0 -4.5254e-02  3.4873e-02 -1.2977 0.1945594    
## arch3                4    0 -2.0508e-02  2.5950e-02 -0.7903 0.4294672    
## arch4                5    0 -5.8071e-03  2.6551e-02 -0.2187 0.8269008    
## arch5                6    0  1.8314e-02  2.6275e-02  0.6970 0.4858772    
## arch6                7    0  9.2361e-02  2.6384e-02  3.5007 0.0004755 ***
## arch7                8    0  1.1531e-01  2.7207e-02  4.2383 2.367e-05 ***
## arch8                9    0  2.4652e-02  2.4320e-02  1.0137 0.3108753    
## arch9               10    0 -3.4962e-02  2.4402e-02 -1.4327 0.1521057    
## arch10              11    0  2.3071e-02  2.4269e-02  0.9506 0.3419114    
## arch11              12    0 -4.9334e-02  2.4206e-02 -2.0381 0.0416843 *  
## arch12              13    0 -4.4547e-02  2.4239e-02 -1.8378 0.0662560 .  
## arch13              14    0  2.0042e-02  2.4108e-02  0.8314 0.4058776    
## arch14              15    0  7.0818e-02  2.3395e-02  3.0271 0.0025047 ** 
## logEqWMA(2)         16    0  2.0253e-01  8.3610e-02  2.4223 0.0155196 *  
## logEqWMA(5)         17    0  1.2332e-01  1.5952e-01  0.7731 0.4395699    
## logEqWMA(7)         18    0  2.0222e-01  1.9212e-01  1.0526 0.2926653    
## logEqWMA(14)        19    0  6.1481e-02  1.2551e-01  0.4899 0.6242875    
## dk2_load_fcast      20    0 -6.0738e-04  2.3565e-04 -2.5775 0.0100320 *  
## dk2_wind_fcast      21    0  9.9166e-04  2.2613e-04  4.3854 1.225e-05 ***
## dk2_se4             22    0 -4.1942e-06  2.3542e-04 -0.0178 0.9857876    
## dk2_se4_2           23    0 -7.3489e-04  1.5501e-04 -4.7410 2.295e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Diagnostics:
## 
##                     Chi-sq df p-value
## Ljung-Box AR(1)    488.750  1 0.00000
## Ljung-Box ARCH(15)  12.608 15 0.63254

## 17 path(s) to search

## Searching: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

## 
##   Path 1: 3 22 5 19 17 4 14 11 6 9 10 13 16 20 12 -15 
##   Path 2: 4 22 5 17 19 14 6 11 9 10 3 13 16 20 12 -15 
##   Path 3: 5 22 19 6 4 17 14 11 9 10 3 13 16 20 12 -15 
##   Path 4: 6 22 5 19 4 17 14 11 9 10 3 13 16 20 12 -15 
##   Path 5: 9 22 5 19 17 4 14 6 11 10 3 13 16 20 12 -15 
##   Path 6: 10 22 19 5 6 11 4 17 9 14 3 13 16 20 12 -15 
##   Path 7: 11 22 5 4 17 19 14 6 9 10 3 13 16 20 12 -15 
##   Path 8: 12 19 22 5 17 11 4 14 6 9 10 3 13 16 20 -15 
##   Path 9: 13 22 19 5 6 14 17 4 11 9 10 3 16 20 12 -15 
##   Path 10: 14 22 5 19 17 4 6 11 9 10 3 13 16 20 12 -15 
##   Path 11: 15 5 22 17 4 11 6 9 10 19 13 3 14 12 16 -20 
##   Path 12: 16 22 6 19 3 5 14 11 10 4 17 9 13 20 12 -15 
##   Path 13: 17 5 22 19 4 14 6 11 9 10 3 13 16 20 12 -15 
##   Path 14: 18 22 5 6 14 11 4 9 3 10 19 13 16 20 12 
##   Path 15: 19 22 5 6 4 17 14 11 9 10 3 13 16 20 12 -15 
##   Path 16: 20 19 5 22 17 4 14 11 6 9 10 3 13 16 12 -15 
##   Path 17: 22 5 19 6 4 17 14 11 9 10 3 13 16 20 12 -15 
## 
## Terminal models:
## 
##         info(sc)      logl    n k
## spec 1: 7.934909 -7159.019 1812 8
## spec 2: 7.971088 -7191.797 1812 8
## spec 3: 7.951754 -7174.280 1812 8
## 
## Retained regressors (final model):
## 
##   vconst   arch1   arch6   arch7   arch14   logEqWMA(7)   dk2_wind_fcast   dk2_se4_2

daily_dk2_vmod %>% plot

Replicate Intraday

Pulled from replication data available at:https://doi-org.ezproxy-prd.bodleian.ox.ac.uk/10.1016/j.eneco.2016.12.019

# Required including "here("code/rintamaki_replication/data/" to data read commands in the listed files 
source(here("code/rintamaki_replication/models/read_intraday_data.R"))

Models as in Rintamäki et al.

intraday_forms <- list(
  list(price_op1_dk1, cbind(wind_op1_dk1, exim_op1_dk1), 2, 1, 1, 1, 7),
  list(price_p_dk1, cbind(wind_p_dk1, exim_p_dk1), 2, 1, 1, 1, 7),
  list(price_op2_dk1, cbind(wind_op2_dk1, exim_op2_dk1), 2, 1, 1, 1, 7),
  list(price_p_dk2, cbind(wind_p_dk2, exim_p_dk2),  2, 1, 1, 1, 7),
  list(price_op2_dk2, cbind(wind_op2_dk2, exim_op2_dk2),  2, 1, 1, 1, 7),
  list(price_op1_de, cbind(wind_op1_de, exim_op1_de),  2, 1, 1, 1, 7),
  list(price_p_de, cbind(wind_p_de, exim_p_de), 2, 1, 1, 1, 7),
  list(price_op2_de, cbind(wind_op2_de, exim_op2_de),  2, 1, 1, 1, 7),
  list(fwd1(price_p_de), cbind(diff(solar_p_de), fwd1(exim_p_de)),  2, 1, 1, 1, 7), 
  list(price_p_de, cbind(res_p_de, exim_p_de), 2, 1, 1, 1, 7))


intraday_res <- lapply(intraday_forms, function(x) do.call(fitModel, x))

modelsummary(intraday_res)

	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
ar1	1.151	1.173	1.062	1.243	1.071	−0.556	0.973	0.591	0.955	0.882
	(0.042)	(0.042)	(0.039)	(0.046)	(0.038)	(0.068)	(0.100)	(0.145)	(0.151)	(0.129)
ar2	−0.192	−0.224	−0.093	−0.291	−0.107	0.295	−0.119	0.099	−0.139	−0.080
	(0.034)	(0.034)	(0.035)	(0.038)	(0.034)	(0.043)	(0.054)	(0.065)	(0.077)	(0.055)
ma1	−0.791	−0.762	−0.722	−0.750	−0.721	0.922	−0.674	−0.343	−0.602	−0.645
	(0.031)	(0.033)	(0.029)	(0.037)	(0.029)	(0.058)	(0.093)	(0.145)	(0.145)	(0.124)
sar1	0.948	0.954	0.970	0.961	0.963	0.917	0.934	0.961	0.928	0.932
	(0.020)	(0.011)	(0.013)	(0.010)	(0.014)	(0.023)	(0.013)	(0.013)	(0.015)	(0.014)
sma1	−0.879	−0.799	−0.936	−0.791	−0.919	−0.734	−0.591	−0.817	−0.609	−0.566
	(0.031)	(0.024)	(0.019)	(0.023)	(0.021)	(0.041)	(0.033)	(0.027)	(0.036)	(0.036)
intercept	4.147	4.169	4.041	4.076	3.930	5.879	4.981	5.172	3.704	7.037
	(0.096)	(0.084)	(0.077)	(0.112)	(0.075)	(0.138)	(0.113)	(0.081)	(0.080)	(0.161)
wind_op1_dk1	−0.109
	(0.009)
exim_op1_dk1	−0.145
	(0.018)
wind_p_dk1		−0.073
		(0.005)
exim_p_dk1		−0.107
		(0.009)
wind_op2_dk1			−0.065
			(0.005)
exim_op2_dk1			−0.087
			(0.009)
wind_p_dk2				−0.057
				(0.004)
exim_p_dk2				−0.066
				(0.012)
wind_op2_dk2					−0.054
					(0.005)
exim_op2_dk2					−0.047
					(0.012)
wind_op1_de						−0.307
						(0.016)
exim_op1_de						0.044
						(0.011)
wind_p_de							−0.153
							(0.009)
exim_p_de							0.019			0.028
							(0.007)			(0.007)
wind_op2_de								−0.187
								(0.008)
exim_op2_de								0.044
								(0.006)
ext1									−0.053
									(0.014)
ext2									0.010
									(0.008)
res_p_de										−0.360
										(0.016)
Num.Obs.	1813	1813	1813	1816	1816	1081	1081	1081	1080	1081
AIC	794.5	−1195.8	−1329.9	−1031.8	−840.6	603.6	−221.5	−827.0	−0.6	−377.6
BIC	844.0	−1146.3	−1280.4	−982.2	−791.1	648.5	−176.6	−782.2	44.3	−332.7
RMSE	0.30	0.17	0.17	0.18	0.19	0.32	0.22	0.16	0.24	0.20

Intraday price model with gets

Data

for(cty in c("de","dk1", "dk2")){
  temp <- read.table(here(paste0("code/rintamaki_replication/data/", cty, "_intraday.txt")), header=TRUE) %>% 
  tibble %>% 
  mutate(date = dmy(date)) %>% 
  arrange(date)
  
  assign(paste0("intraday_data_", cty), temp)
}

Germany

intraday_data_de %>% 
  select(date, op1_price, op2_price, p_price) %>% 
  pivot_longer(cols = !date) %>% 
  ggplot() +
  geom_line(aes(x = date, y = value, color = name), linewidth = 0.1) +
  labs(main = "DE: Intraday Electricity Prices")

intraday_data_de %>% arrange(date) %>% identical(intraday_data_de)

## [1] TRUE

for(dep in c("op1_", "op2_", "p_")){
  temp <- intraday_data_de %>% 
    select(date, contains(dep))
  print(temp)
  
  dep2 <- as.matrix(temp[paste0(dep, "price")])
  vxregs <- as.matrix(temp[!(names(temp) %in% c("date", paste0(dep, "price")))])

  temp_mod <- arx(dep2, 
              ar = 1:14, 
              vxreg = vxregs,
              log.ewma = c(2, 5, 7, 14)) 
  assign(paste0("de_", dep, "mod"), temp_mod)
  
  temp_vmod <- getsv(temp_mod, t.pval = 0.001, ar.LjungB = NULL, arch.LjungB = NULL)
  assign(paste0("de_", dep, "vmod"), temp_vmod)

  plot(temp_vmod)
  
}

## # A tibble: 1,081 × 8
##    date       op1_price op1_load op1_wind op1_res op1_wind_pen op1_res_pen
##    <date>         <dbl>    <dbl>    <dbl>   <dbl>        <dbl>       <dbl>
##  1 2012-01-01      2.61     10.5     8.74    8.74       -1.71       -1.71 
##  2 2012-01-02      2.08     10.5     9.34    9.34       -1.16       -1.16 
##  3 2012-01-03      3.34     10.7     9.29    9.29       -1.45       -1.45 
##  4 2012-01-04      2.38     10.7     9.95    9.95       -0.779      -0.779
##  5 2012-01-06      2.76     10.7     9.78    9.78       -0.934      -0.934
##  6 2012-01-07      3.14     10.7     9.47    9.47       -1.19       -1.19 
##  7 2012-01-08      1.27     10.6     9.55    9.55       -1.01       -1.01 
##  8 2012-01-09      3.64     10.8     8.13    8.13       -2.62       -2.62 
##  9 2012-01-10      3.53     10.8     9.28    9.28       -1.48       -1.48 
## 10 2012-01-11      3.63     10.8     9.03    9.03       -1.80       -1.80 
## # ℹ 1,071 more rows
## # ℹ 1 more variable: op1_exp <dbl>
## GUM log-variance equation:
## 
##              reg.no. keep      coef std.error  t-stat   p-value    
## vconst             1    1 26.983190  7.591860 12.6325 0.0003791 ***
## logEqWMA(2)        2    0  0.262951  0.055182  4.7651 2.155e-06 ***
## logEqWMA(5)        3    0 -0.460613  0.144093 -3.1966 0.0014322 ** 
## logEqWMA(7)        4    0  0.597089  0.174600  3.4197 0.0006511 ***
## logEqWMA(14)       5    0  0.170359  0.121342  1.4040 0.1606291    
## op1_load           6    0  0.906020  4.571494  0.1982 0.8429358    
## op1_wind           7    0 -9.622936  9.422003 -1.0213 0.3073371    
## op1_res            8    0  6.194747  9.805404  0.6318 0.5276766    
## op1_wind_pen       9    0 10.259600 11.019319  0.9311 0.3520404    
## op1_res_pen       10    0 -6.413083 11.510264 -0.5572 0.5775363    
## op1_exp           11    0  0.186567  0.056144  3.3230 0.0009215 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Diagnostics:
## 
##                     Chi-sq df  p-value
## Ljung-Box AR(15)  23.17148 15 0.080566
## Ljung-Box ARCH(1)  0.33314  1 0.563818

## 7 path(s) to search

## Searching: 1 2 3 4 5 6 7

## 
##   Path 1: 3 6 10 8 4 -11 
##   Path 2: 5 6 10 8 3 
##   Path 3: 6 10 8 5 3 
##   Path 4: 7 6 9 5 
##   Path 5: 8 6 10 5 3 
##   Path 6: 9 6 7 5 
##   Path 7: 10 6 8 5 3 
## 
## Terminal models:
## 
##          info(sc)      logl    n k
## spec 1: 0.1695989 -68.41564 1053 6
## spec 2: 0.1832057 -75.57962 1053 6
## spec 3: 0.1950007 -78.30997 1053 7
## 
## Retained regressors (final model):
## 
##   vconst   logEqWMA(2)   logEqWMA(14)   op1_wind   op1_wind_pen   op1_exp

## # A tibble: 1,081 × 8
##    date       op2_price op2_load op2_wind op2_res op2_wind_pen op2_res_pen
##    <date>         <dbl>    <dbl>    <dbl>   <dbl>        <dbl>       <dbl>
##  1 2012-01-01      3.24     10.7     9.35    9.35       -1.31       -1.31 
##  2 2012-01-02      3.62     10.9     9.22    9.22       -1.67       -1.67 
##  3 2012-01-03      3.30     10.9    10.0    10.0        -0.861      -0.861
##  4 2012-01-04      3.40     10.9    10.0    10.0        -0.906      -0.906
##  5 2012-01-06      3.78     10.9     9.15    9.15       -1.73       -1.73 
##  6 2012-01-07      3.43     10.8     9.60    9.60       -1.20       -1.20 
##  7 2012-01-08      3.74     10.8     8.50    8.50       -2.32       -2.32 
##  8 2012-01-09      3.77     11.0     9.10    9.10       -1.85       -1.85 
##  9 2012-01-10      3.77     11.0     8.77    8.77       -2.21       -2.21 
## 10 2012-01-11      3.88     11.0     9.01    9.01       -1.96       -1.96 
## # ℹ 1,071 more rows
## # ℹ 1 more variable: op2_exp <dbl>
## GUM log-variance equation:
## 
##              reg.no. keep       coef  std.error  t-stat  p-value   
## vconst             1    1 -22.756861   7.590465  8.9885 0.002717 **
## logEqWMA(2)        2    0   0.080081   0.062014  1.2913 0.196874   
## logEqWMA(5)        3    0   0.059842   0.164864  0.3630 0.716694   
## logEqWMA(7)        4    0   0.248518   0.200493  1.2395 0.215426   
## logEqWMA(14)       5    0   0.073629   0.142228  0.5177 0.604791   
## op2_load           6    0  11.749422  12.436521  0.9448 0.345005   
## op2_wind           7    0  45.681518  59.430292  0.7687 0.442271   
## op2_res            8    0 -55.346728  62.659197 -0.8833 0.377279   
## op2_wind_pen       9    0 -47.386760  64.435596 -0.7354 0.462254   
## op2_res_pen       10    0  57.618481  67.761896  0.8503 0.395349   
## op2_exp           11    0  -0.128795   0.064903 -1.9844 0.047471 * 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Diagnostics:
## 
##                      Chi-sq df p-value
## Ljung-Box AR(15)  16.099244 15 0.37549
## Ljung-Box ARCH(1)  0.000123  1 0.99115

## 10 path(s) to search
## Searching: 1 2 3 4 5 6 7 8 9 10

## 
##   Path 1: 2 5 9 10 3 8 11 6 
##   Path 2: 3 5 9 10 8 2 11 6 
##   Path 3: 4 9 10 8 2 5 11 6 
##   Path 4: 5 3 9 10 8 2 11 6 
##   Path 5: 6 3 5 10 8 2 11 9 
##   Path 6: 7 3 5 10 8 2 11 6 
##   Path 7: 8 3 5 7 10 2 11 6 
##   Path 8: 9 3 5 10 8 2 11 6 
##   Path 9: 10 7 3 5 8 2 11 6 
##   Path 10: 11 5 3 7 9 8 2 6 
## 
## Terminal models:
## 
##           info(sc)     logl    n k
## spec 1: -0.7303577 394.9725 1053 3
## spec 2: -0.7303367 394.9614 1053 3
## spec 3: -0.7252047 392.2593 1053 3
## spec 4: -0.7236655 391.4490 1053 3
## 
## Retained regressors (final model):
## 
##   vconst   logEqWMA(7)   op2_wind

## # A tibble: 1,081 × 10
##    date       p_price p_load p_wind p_res p_wind_pen p_res_pen  p_exp p_solar
##    <date>       <dbl>  <dbl>  <dbl> <dbl>      <dbl>     <dbl>  <dbl>   <dbl>
##  1 2012-01-01    2.93   10.6   9.21  9.25      -1.41     -1.37 -0.716    5.88
##  2 2012-01-02    3.91   11.0   9.07  9.13      -1.96     -1.90 -1.8      6.21
##  3 2012-01-03    3.74   11.0   9.94 10.0       -1.10     -1.04 -1.23     7.23
##  4 2012-01-04    3.85   11.0   9.96 10.0       -1.09     -1.04  0.296    7.05
##  5 2012-01-06    3.86   11.0   9.31  9.41      -1.68     -1.58  0.207    7.06
##  6 2012-01-07    3.68   10.9   9.63  9.65      -1.25     -1.23  0.662    5.59
##  7 2012-01-08    3.62   10.8   9.16  9.22      -1.63     -1.57  0.102    6.25
##  8 2012-01-09    4.03   11.1   8.60  8.68      -2.54     -2.47 -0.999    6.06
##  9 2012-01-10    3.99   11.1   8.70  8.86      -2.41     -2.25 -0.548    6.97
## 10 2012-01-11    3.99   11.1   9.12  9.22      -1.99     -1.90  0.992    6.82
## # ℹ 1,071 more rows
## # ℹ 1 more variable: p_solar_pen <dbl>
## GUM log-variance equation:
## 
##              reg.no. keep       coef  std.error  t-stat p-value  
## vconst             1    1   9.056373   9.228539  0.9630 0.32642  
## logEqWMA(2)        2    0   0.110686   0.059891  1.8481 0.06487 .
## logEqWMA(5)        3    0  -0.286182   0.154388 -1.8537 0.06407 .
## logEqWMA(7)        4    0   0.379753   0.195660  1.9409 0.05254 .
## logEqWMA(14)       5    0   0.225599   0.160425  1.4063 0.15994  
## p_load             6    0 -14.783766  19.062209 -0.7756 0.43819  
## p_wind             7    0  31.713581  14.406761  2.2013 0.02793 *
## p_res              8    0  -3.620176  30.150302 -0.1201 0.90445  
## p_wind_pen         9    0 -31.650357  14.478575 -2.1860 0.02904 *
## p_res_pen         10    0   3.907697  30.253330  0.1292 0.89725  
## p_exp             11    0  -0.078726   0.062948 -1.2506 0.21134  
## p_solar           12    0 -14.221736  18.835651 -0.7550 0.45039  
## p_solar_pen       13    0  14.115591  18.880614  0.7476 0.45486  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Diagnostics:
## 
##                   Chi-sq df    p-value
## Ljung-Box AR(15)  18.724 15 0.22653628
## Ljung-Box ARCH(1) 13.955  1 0.00018726

## 12 path(s) to search
## Searching: 1 2 3 4 5 6 7 8 9 10 11 12

## 
##   Path 1: 2 8 10 6 3 5 11 9 7 
##   Path 2: 3 8 10 6 4 11 2 9 7 
##   Path 3: 4 8 3 10 6 11 2 9 7 
##   Path 4: 5 8 10 6 11 2 3 9 7 
##   Path 5: 6 8 10 5 11 2 3 9 7 
##   Path 6: 7 6 11 9 5 10 2 3 8 
##   Path 7: 8 10 6 5 11 2 3 9 7 
##   Path 8: 9 6 11 7 5 10 2 3 8 
##   Path 9: 10 8 6 5 11 2 3 9 7 
##   Path 10: 11 8 6 10 5 2 3 9 7 
##   Path 11: 12 6 13 5 11 2 3 9 7 -8 -10 
##   Path 12: 13 6 12 5 11 2 3 9 7 -8 -10 
## 
## Terminal models:
## 
##            info(sc)     logl    n k
## spec 1: -0.06084853 45.95555 1053 4
## spec 2: -0.07041061 50.98998 1053 4
## spec 3: -0.05658669 43.71169 1053 4
## 
## Retained regressors (final model):
## 
##   vconst   logEqWMA(14)   p_solar   p_solar_pen

DK1

intraday_data_dk1 %>% 
  select(date, op1_dk1_price, op2_dk1_price, p_dk1_price) %>% 
  pivot_longer(cols = !date) %>% 
  ggplot() +
  geom_line(aes(x = date, y = value, color = name), linewidth = 0.1) +
  labs(main = "DK1: Intraday Electricity Prices")

intraday_data_dk1 %>% arrange(date) %>% identical(intraday_data_dk1)

## [1] TRUE

for(dep in c("op1_dk1_", "op2_dk1_", "p_dk1_")){
  temp <- intraday_data_dk1 %>% 
    select(date, contains(dep))
  print(temp)
  
  dep2 <- as.matrix(temp[paste0(dep, "price")])
  vxregs <- as.matrix(temp[!(names(temp) %in% c("date", paste0(dep, "price")))])

  temp_mod <- arx(dep2, 
              arch = 1:14, 
              vxreg = vxregs,
              log.ewma = c(2, 5, 7, 14)) 
  assign(paste0("dk1_", dep, "mod"), temp_mod)
  
  temp_vmod <- getsv(temp_mod, t.pval = 0.001, ar.LjungB = NULL, arch.LjungB = NULL)
  assign(paste0("dk1_", dep, "vmod"), temp_vmod)

  plot(temp_vmod)
  
}

## # A tibble: 1,813 × 6
##    date       op1_dk1_price op1_dk1_load op1_dk1_wind op1_dk1_pen op1_dk1_exp
##    <date>             <dbl>        <dbl>        <dbl>       <dbl>       <dbl>
##  1 2010-01-01          3.40         7.51         5.52     -1.98          1.36
##  2 2010-01-02          3.64         7.53         6.13     -1.40          1.32
##  3 2010-01-03          3.63         7.45         5.04     -2.40          1.26
##  4 2010-01-04          3.57         7.70         5.62     -2.06          1.20
##  5 2010-01-05          3.72         7.78         5.05     -2.73          1.22
##  6 2010-01-06          3.59         7.71         6.80     -0.885         1.31
##  7 2010-01-07          3.65         7.71         6.16     -1.52          1.20
##  8 2010-01-08          3.73         7.73         5.69     -2.04          1.07
##  9 2010-01-09          3.27         7.53         6.65     -0.884         1.37
## 10 2010-01-10          2.71         7.59         7.53     -0.0543        1.37
## # ℹ 1,803 more rows
## GUM log-variance equation:
## 
##              reg.no. keep        coef   std.error  t-stat   p-value    
## vconst             1    1  3.94967009  3.13326383  1.5890  0.207467    
## arch1              2    0  0.16701611  0.03691295  4.5246 6.452e-06 ***
## arch2              3    0  0.00180207  0.03611419  0.0499  0.960208    
## arch3              4    0  0.02867221  0.02595641  1.1046  0.269470    
## arch4              5    0  0.07832347  0.02640039  2.9668  0.003050 ** 
## arch5              6    0  0.04757932  0.02658310  1.7898  0.073651 .  
## arch6              7    0  0.07909038  0.02706117  2.9227  0.003515 ** 
## arch7              8    0  0.11936467  0.02693678  4.4313 9.939e-06 ***
## arch8              9    0  0.06013641  0.02471535  2.4332  0.015066 *  
## arch9             10    0  0.00384697  0.02480626  0.1551  0.876775    
## arch10            11    0 -0.00663179  0.02477847 -0.2676  0.789005    
## arch11            12    0  0.00847630  0.02474596  0.3425  0.731991    
## arch12            13    0 -0.01645513  0.02461777 -0.6684  0.503949    
## arch13            14    0 -0.00074796  0.02470069 -0.0303  0.975846    
## arch14            15    0  0.04897218  0.02402341  2.0385  0.041646 *  
## logEqWMA(2)       16    0  0.19734151  0.07679945  2.5696  0.010263 *  
## logEqWMA(5)       17    0  0.02446353  0.11852230  0.2064  0.836499    
## logEqWMA(7)       18    0 -0.11265405  0.13474672 -0.8360  0.403243    
## logEqWMA(14)      19    0  0.00037131  0.09841278  0.0038  0.996990    
## op1_dk1_load      20    0 -2.16224468  2.31871110 -0.9325  0.351195    
## op1_dk1_wind      21    0  1.73247930  2.42376285  0.7148  0.474833    
## op1_dk1_pen       22    0 -1.74817615  2.41868056 -0.7228  0.469910    
## op1_dk1_exp       23    0  0.35441960  0.06558171  5.4042 7.388e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Diagnostics:
## 
##                      Chi-sq df p-value
## Ljung-Box AR(1)    242.0837  1 0.00000
## Ljung-Box ARCH(15)   1.8586 15 0.99998

## 19 path(s) to search

## Searching: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

## 
##   Path 1: 3 19 14 10 17 11 12 13 21 22 4 18 20 6 15 9 
##   Path 2: 4 14 19 11 10 3 12 17 13 21 22 18 20 6 15 9 
##   Path 3: 5 14 11 19 10 3 13 12 21 22 17 18 20 4 15 6 
##   Path 4: 6 14 19 3 11 10 12 13 21 22 17 18 4 20 15 9 
##   Path 5: 7 11 3 10 19 14 18 12 13 21 22 17 4 20 15 6 9 
##   Path 6: 9 14 3 17 11 12 19 13 21 22 10 20 4 18 6 15 
##   Path 7: 10 14 19 3 17 11 12 13 21 22 4 18 20 6 15 9 
##   Path 8: 11 14 3 19 10 17 12 13 21 22 4 18 20 6 15 9 
##   Path 9: 12 14 3 19 10 11 17 13 21 22 4 18 20 6 15 9 
##   Path 10: 13 3 19 10 17 14 12 11 21 22 4 18 20 6 15 9 
##   Path 11: 14 19 3 10 17 11 12 13 21 22 4 18 20 6 15 9 
##   Path 12: 15 3 17 10 11 12 14 13 19 21 22 20 4 18 6 9 
##   Path 13: 16 19 14 11 12 10 21 22 13 4 18 17 20 6 15 9 
##   Path 14: 17 19 14 3 10 11 12 13 21 22 4 18 20 6 15 9 
##   Path 15: 18 3 14 11 10 12 19 13 21 22 17 4 20 6 15 9 
##   Path 16: 19 14 3 10 17 11 12 13 21 22 4 18 20 6 15 9 
##   Path 17: 20 14 19 3 17 10 11 12 13 22 21 4 18 6 15 9 
##   Path 18: 21 14 19 3 10 17 11 12 22 13 4 18 20 6 15 9 
##   Path 19: 22 14 19 3 10 17 11 12 21 13 4 18 20 6 15 9 
## 
## Terminal models:
## 
##         info(sc)       logl    n k
## spec 1: 1.137488  -996.9384 1799 7
## spec 2: 1.066120  -932.7429 1799 7
## spec 3: 1.155852 -1017.2036 1799 6
## spec 4: 1.245907 -1094.4610 1799 7
## 
## Retained regressors (final model):
## 
##   vconst   arch1   arch6   arch7   arch8   logEqWMA(2)   op1_dk1_exp

## # A tibble: 1,813 × 6
##    date       op2_dk1_price op2_dk1_load op2_dk1_wind op2_dk1_pen op2_dk1_exp
##    <date>             <dbl>        <dbl>        <dbl>       <dbl>       <dbl>
##  1 2010-01-01          3.66         7.67         5.91      -1.76        1.37 
##  2 2010-01-02          3.75         7.68         6.09      -1.60       -0.28 
##  3 2010-01-03          3.70         7.72         5.18      -2.52        0.567
##  4 2010-01-04          3.73         7.91         5.57      -2.32        1.07 
##  5 2010-01-05          3.67         7.93         7.05      -0.863       1.15 
##  6 2010-01-06          3.85         7.87         6.42      -1.45       -0.393
##  7 2010-01-07          3.85         7.87         5.41      -2.45        0.223
##  8 2010-01-08          3.67         7.89         6.44      -1.44        1.37 
##  9 2010-01-09          3.69         7.69         7.39      -0.301       1.37 
## 10 2010-01-10          3.78         7.67         7.33      -0.329       1.34 
## # ℹ 1,803 more rows
## GUM log-variance equation:
## 
##              reg.no. keep       coef  std.error  t-stat   p-value    
## vconst             1    1  7.8498660  3.2599679  5.7983  0.016042 *  
## arch1              2    0  0.1597272  0.0355229  4.4965 7.357e-06 ***
## arch2              3    0 -0.0584432  0.0360218 -1.6224  0.104887    
## arch3              4    0  0.0039254  0.0264632  0.1483  0.882095    
## arch4              5    0  0.0566032  0.0269395  2.1011  0.035771 *  
## arch5              6    0  0.0237831  0.0271404  0.8763  0.380985    
## arch6              7    0  0.0929221  0.0269126  3.4527  0.000568 ***
## arch7              8    0  0.1727698  0.0273475  6.3176 3.351e-10 ***
## arch8              9    0  0.0247232  0.0249317  0.9916  0.321510    
## arch9             10    0  0.0070094  0.0248114  0.2825  0.777586    
## arch10            11    0 -0.0068903  0.0248354 -0.2774  0.781476    
## arch11            12    0 -0.0014211  0.0247246 -0.0575  0.954170    
## arch12            13    0 -0.0301110  0.0247153 -1.2183  0.223267    
## arch13            14    0  0.0227127  0.0247411  0.9180  0.358737    
## arch14            15    0  0.0565874  0.0238666  2.3710  0.017847 *  
## logEqWMA(2)       16    0  0.2439252  0.0795694  3.0656  0.002205 ** 
## logEqWMA(5)       17    0  0.3357595  0.1364717  2.4603  0.013977 *  
## logEqWMA(7)       18    0 -0.1756809  0.1507023 -1.1657  0.243873    
## logEqWMA(14)      19    0 -0.0944631  0.1038504 -0.9096  0.363153    
## op2_dk1_load      20    0 -0.1811236  4.5993200 -0.0394  0.968591    
## op2_dk1_wind      21    0 -0.7718292  4.6806804 -0.1649  0.869044    
## op2_dk1_pen       22    0  0.7721362  4.6782446  0.1650  0.868925    
## op2_dk1_exp       23    0  0.3234266  0.0516240  6.2650 4.666e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Diagnostics:
## 
##                       Chi-sq df p-value
## Ljung-Box AR(1)    362.36474  1       0
## Ljung-Box ARCH(15)   0.65534 15       1

## 18 path(s) to search
## Searching: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

## 
##   Path 1: 3 4 20 12 10 11 6 9 19 14 13 18 5 22 21 16 15 
##   Path 2: 4 20 12 11 10 6 14 19 9 13 3 18 5 22 21 16 15 
##   Path 3: 5 20 11 4 12 10 6 14 19 9 13 3 18 22 21 16 15 
##   Path 4: 6 12 4 20 11 10 14 19 9 13 3 18 5 22 21 16 15 
##   Path 5: 9 20 12 4 11 10 19 14 6 13 3 18 5 22 21 16 15 
##   Path 6: 10 20 12 4 11 6 14 19 9 13 3 18 5 22 21 16 15 
##   Path 7: 11 20 4 12 10 6 14 19 9 13 3 18 5 22 21 16 15 
##   Path 8: 12 20 4 11 10 6 14 19 9 13 3 18 5 22 21 16 15 
##   Path 9: 13 20 4 11 10 12 6 14 9 19 3 18 5 22 21 16 15 
##   Path 10: 14 20 12 4 11 10 19 9 6 13 3 18 5 22 21 16 15 
##   Path 11: 15 20 12 4 11 10 19 6 9 13 14 3 18 5 22 21 16 
##   Path 12: 16 12 20 4 10 6 11 3 14 19 9 13 5 18 22 21 15 
##   Path 13: 17 20 12 10 11 4 9 14 19 18 13 3 6 22 21 15 
##   Path 14: 18 20 12 11 4 10 6 13 14 9 3 19 5 22 21 16 15 
##   Path 15: 19 20 10 4 12 11 14 9 6 13 3 18 5 22 21 16 15 
##   Path 16: 20 12 4 11 10 6 14 19 9 13 3 18 5 22 21 16 15 
##   Path 17: 21 22 12 4 11 10 6 14 19 9 13 3 18 5 20 16 15 
##   Path 18: 22 21 12 4 11 10 6 14 19 9 13 3 18 5 20 16 15 
## 
## Terminal models:
## 
##          info(sc)      logl    n k
## spec 1: 0.1895942 -148.0550 1799 6
## spec 2: 0.1651644 -122.3329 1799 7
## 
## Retained regressors (final model):
## 
##   vconst   arch1   arch4   arch6   arch7   logEqWMA(2)   op2_dk1_exp

## # A tibble: 1,813 × 6
##    date       p_dk1_price p_dk1_load p_dk1_wind p_dk1_pen p_dk1_exp
##    <date>           <dbl>      <dbl>      <dbl>     <dbl>     <dbl>
##  1 2010-01-01        3.62       7.70       6.07    -1.62     1.36  
##  2 2010-01-02        3.79       7.82       6.59    -1.23     0.0989
##  3 2010-01-03        3.77       7.78       4.75    -3.05     0.431 
##  4 2010-01-04        3.83       8.11       5.46    -2.65     0.351 
##  5 2010-01-05        3.93       8.16       6.57    -1.58     0.186 
##  6 2010-01-06        3.90       8.09       6.66    -1.43     0.375 
##  7 2010-01-07        4.10       8.09       5.77    -2.31     0.407 
##  8 2010-01-08        3.96       8.11       6.30    -1.80     0.689 
##  9 2010-01-09        3.74       7.85       7.19    -0.661    1.16  
## 10 2010-01-10        3.61       7.83       7.51    -0.325    1.26  
## # ℹ 1,803 more rows
## GUM log-variance equation:
## 
##              reg.no. keep      coef std.error  t-stat   p-value    
## vconst             1    1  1.748866  2.211920  0.6251  0.429145    
## arch1              2    0  0.175281  0.036380  4.8181 1.573e-06 ***
## arch2              3    0 -0.037836  0.037969 -0.9965  0.319147    
## arch3              4    0 -0.039307  0.026388 -1.4896  0.136502    
## arch4              5    0  0.015597  0.027325  0.5708  0.568224    
## arch5              6    0  0.002618  0.027008  0.0969  0.922789    
## arch6              7    0  0.074790  0.026980  2.7721  0.005628 ** 
## arch7              8    0  0.152106  0.027558  5.5195 3.902e-08 ***
## arch8              9    0  0.034448  0.024775  1.3905  0.164565    
## arch9             10    0 -0.034748  0.024754 -1.4038  0.160563    
## arch10            11    0  0.024846  0.024717  1.0052  0.314932    
## arch11            12    0  0.022674  0.024652  0.9198  0.357821    
## arch12            13    0 -0.002436  0.024769 -0.0984  0.921665    
## arch13            14    0  0.019134  0.024835  0.7704  0.441153    
## arch14            15    0  0.067276  0.024183  2.7820  0.005459 ** 
## logEqWMA(2)       16    0  0.172992  0.084123  2.0564  0.039888 *  
## logEqWMA(5)       17    0  0.055213  0.143303  0.3853  0.700072    
## logEqWMA(7)       18    0  0.104332  0.165657  0.6298  0.528902    
## logEqWMA(14)      19    0 -0.104788  0.123026 -0.8518  0.394466    
## p_dk1_load        20    0 -7.042586  4.245288 -1.6589  0.097309 .  
## p_dk1_wind        21    0  6.824227  4.221929  1.6164  0.106191    
## p_dk1_pen         22    0 -6.839021  4.225353 -1.6186  0.105718    
## p_dk1_exp         23    0  0.253469  0.054342  4.6643 3.327e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Diagnostics:
## 
##                     Chi-sq df  p-value
## Ljung-Box AR(1)    394.799  1 0.000000
## Ljung-Box ARCH(15)  30.523 15 0.010168

## 19 path(s) to search
## Searching: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

## 
##   Path 1: 3 6 13 5 18 19 14 11 12 9 10 4 17 21 22 20 15 16 
##   Path 2: 4 17 13 6 5 14 19 11 3 12 9 10 18 21 22 20 15 16 
##   Path 3: 5 13 6 18 14 19 11 3 12 9 10 4 17 21 22 20 15 16 
##   Path 4: 6 13 17 14 19 12 3 5 11 4 9 10 18 21 22 20 15 16 
##   Path 5: 7 13 17 6 5 3 11 19 14 12 9 10 21 22 20 16 4 15 
##   Path 6: 9 6 13 17 19 14 12 3 5 11 10 4 18 21 22 20 15 16 
##   Path 7: 10 13 6 17 11 14 5 19 9 3 12 4 18 21 22 20 15 16 
##   Path 8: 11 6 13 17 14 19 5 3 12 4 9 10 18 21 22 20 15 16 
##   Path 9: 12 13 6 17 19 14 3 5 11 4 9 10 18 21 22 20 15 16 
##   Path 10: 13 6 17 14 19 12 3 5 11 4 9 10 18 21 22 20 15 16 
##   Path 11: 14 13 6 17 19 12 3 5 11 4 9 10 18 21 22 20 15 16 
##   Path 12: 15 6 13 19 18 5 12 3 11 4 14 17 10 9 21 22 20 16 
##   Path 13: 16 5 13 6 18 14 19 11 3 12 10 9 4 21 22 20 17 15 
##   Path 14: 17 13 6 14 19 12 3 5 11 4 9 10 18 21 22 20 15 16 
##   Path 15: 18 6 13 5 14 19 11 3 12 9 10 4 17 21 22 20 15 16 
##   Path 16: 19 6 18 13 14 5 11 3 12 9 10 4 17 21 22 20 15 16 
##   Path 17: 20 6 13 17 22 21 5 14 19 3 11 12 4 9 10 18 15 16 
##   Path 18: 21 6 13 22 17 20 5 14 19 3 11 12 4 9 10 18 15 16 
##   Path 19: 22 6 13 21 17 20 5 14 19 3 11 12 4 9 10 18 15 16 
## 
## Terminal models:
## 
##          info(sc)      logl    n k
## spec 1: 0.4348420 -372.4030 1799 5
## spec 2: 0.3995534 -340.6608 1799 5
## 
## Retained regressors (final model):
## 
##   vconst   arch1   arch7   logEqWMA(7)   p_dk1_exp

DK2

intraday_data_dk2 %>% 
  select(date, op1_dk2_price, op2_dk2_price, p_dk2_price) %>% 
  pivot_longer(cols = !date) %>% 
  ggplot() +
  geom_line(aes(x = date, y = value, color = name), linewidth = 0.1) +
  labs(main = "dk2: Intraday Electricity Prices")

intraday_data_dk2 %>% arrange(date) %>% identical(intraday_data_dk2)

## [1] TRUE

for(dep in c("op1_dk2_", "op2_dk2_", "p_dk2_")){
  temp <- intraday_data_dk2 %>% 
    select(date, contains(dep))
  print(temp)
  
  dep2 <- as.matrix(temp[paste0(dep, "price")])
  vxregs <- as.matrix(temp[!(names(temp) %in% c("date", paste0(dep, "price")))])

  temp_mod <- arx(dep2, 
              arch = 1:14, 
              vxreg = vxregs,
              log.ewma = c(2, 5, 7, 14)) 
  assign(paste0("dk2_", dep, "mod"), temp_mod)
  
  temp_vmod <- getsv(temp_mod, t.pval = 0.001, ar.LjungB = NULL, arch.LjungB = NULL)
  assign(paste0("dk2_", dep, "vmod"), temp_vmod)

  printtex(temp_vmod)
  plot(temp_vmod)
  
}

## # A tibble: 1,816 × 6
##    date       op1_dk2_price op1_dk2_load op1_dk2_wind op1_dk2_pen op1_dk2_exp
##    <date>             <dbl>        <dbl>        <dbl>       <dbl>       <dbl>
##  1 2010-01-01          3.67         7.28         4.82      -2.45        0.497
##  2 2010-01-02          3.68         7.29         4.90      -2.40        0.481
##  3 2010-01-03          3.70         7.25         4.17      -3.07        0.687
##  4 2010-01-04          3.75         7.41         5.30      -2.10        0.884
##  5 2010-01-05          3.76         7.45         4.80      -2.62        0.681
##  6 2010-01-06          3.78         7.42         4.91      -2.48        0.910
##  7 2010-01-07          4.07         7.42         4.64      -2.77        0.813
##  8 2010-01-08          5.15         7.47         4.51      -2.96        1.05 
##  9 2010-01-09          3.83         7.29         6.05      -1.25        1.35 
## 10 2010-01-10          3.81         7.36         6.43      -0.936       1.22 
## # ℹ 1,806 more rows
## GUM log-variance equation:
## 
##              reg.no. keep       coef  std.error  t-stat   p-value    
## vconst             1    1  2.0846581  2.9842730  0.4880 0.4848345    
## arch1              2    0  0.1701187  0.0363692  4.6776 3.122e-06 ***
## arch2              3    0 -0.0418708  0.0364035 -1.1502 0.2502217    
## arch3              4    0  0.0618691  0.0264923  2.3354 0.0196351 *  
## arch4              5    0  0.0872872  0.0269417  3.2399 0.0012180 ** 
## arch5              6    0  0.0497003  0.0271839  1.8283 0.0676715 .  
## arch6              7    0  0.0630632  0.0271463  2.3231 0.0202869 *  
## arch7              8    0  0.1185157  0.0275589  4.3005 1.797e-05 ***
## arch8              9    0  0.0753762  0.0247884  3.0408 0.0023940 ** 
## arch9             10    0 -0.0026942  0.0250186 -0.1077 0.9142544    
## arch10            11    0 -0.0104003  0.0249410 -0.4170 0.6767320    
## arch11            12    0 -0.0288095  0.0249267 -1.1558 0.2479312    
## arch12            13    0  0.0025599  0.0250103  0.1024 0.9184880    
## arch13            14    0 -0.0085318  0.0248995 -0.3426 0.7319040    
## arch14            15    0  0.0602917  0.0240353  2.5085 0.0122142 *  
## logEqWMA(2)       16    0  0.3182909  0.0766389  4.1531 3.435e-05 ***
## logEqWMA(5)       17    0 -0.1671147  0.1224058 -1.3653 0.1723467    
## logEqWMA(7)       18    0  0.0366417  0.1371773  0.2671 0.7894141    
## logEqWMA(14)      19    0  0.0605627  0.0987853  0.6131 0.5399063    
## op1_dk2_load      20    0 -0.5253182  2.2945104 -0.2289 0.8189375    
## op1_dk2_wind      21    0  0.3679733  2.3131041  0.1591 0.8736223    
## op1_dk2_pen       22    0 -0.2957501  2.3070646 -0.1282 0.8980106    
## op1_dk2_exp       23    0  0.2987772  0.0806134  3.7063 0.0002167 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Diagnostics:
## 
##                     Chi-sq df p-value
## Ljung-Box AR(1)    358.765  1 0.00000
## Ljung-Box ARCH(15)  22.058 15 0.10629

## 18 path(s) to search

## Searching: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

## 
##   Path 1: 3 22 13 10 18 14 11 20 19 12 17 6 4 21 15 
##   Path 2: 4 10 13 11 22 18 14 20 19 17 3 12 6 21 15 
##   Path 3: 5 10 18 22 14 13 11 17 19 20 12 3 21 6 4 15 
##   Path 4: 6 10 18 14 13 22 11 20 19 17 3 12 4 21 15 
##   Path 5: 7 10 14 22 13 11 19 20 3 12 21 18 17 4 6 15 
##   Path 6: 9 18 13 22 14 11 10 20 3 19 12 17 6 21 4 15 
##   Path 7: 10 13 22 18 14 11 20 19 3 12 17 6 4 21 15 
##   Path 8: 11 13 22 10 18 14 20 19 3 12 17 6 4 21 15 
##   Path 9: 12 10 13 22 19 14 20 11 18 3 17 6 4 21 15 
##   Path 10: 13 10 22 18 14 11 20 19 3 12 17 6 4 21 15 
##   Path 11: 14 13 10 22 18 11 20 19 3 12 17 6 4 21 15 
##   Path 12: 15 18 22 13 10 14 11 20 3 12 19 17 6 21 4 
##   Path 13: 17 13 10 22 14 11 19 20 18 3 6 12 4 21 15 
##   Path 14: 18 13 10 22 14 11 20 19 3 12 17 6 4 21 15 
##   Path 15: 19 10 22 14 13 11 20 18 3 12 17 6 4 21 15 
##   Path 16: 20 13 10 18 14 21 11 19 3 12 17 6 4 22 15 
##   Path 17: 21 13 10 18 14 20 11 19 3 12 17 6 4 22 15 
##   Path 18: 22 13 10 18 14 11 20 19 3 12 17 6 4 21 15 
## 
## Terminal models:
## 
##          info(sc)      logl    n k
## spec 1: 0.8378096 -724.8798 1802 8
## spec 2: 0.8350453 -726.1375 1802 7
## spec 3: 0.8517737 -741.2098 1802 7
## spec 4: 0.8375691 -728.4114 1802 7
## 
## Retained regressors (final model):
## 
##   vconst   arch1   arch6   arch7   arch8   logEqWMA(2)   op1_dk2_exp 
## % Date: Tue Sep 17 14:26:55 2024 
## % LaTeX code generated in R 4.3.1 by the gets package
## % Note: The {eqnarray} environment requires the {amsmath} package
## \begin{eqnarray}
##   \widehat{e} &=& \underset{(0.0324)}{0.5921} + \underset{(0.0228)}{0.1884}arch1 + \underset{(0.0240)}{0.1006}arch6 + \underset{(0.0233)}{0.1445}arch7 + \underset{(0.0431)}{0.0923}arch8 + \underset{(0.0684)}{0.2567}logEqWMA(2) + \underset{(NA)}{0.3286}op1_dk2_exp - \underset{(NA)}{1.5046}Elnz2 \\[2mm] 
##    && R^2=NA \qquad \widehat{\sigma}=0.4437 \qquad LogL=-726.1375 \qquad T = 1802 \nonumber \\ 
## \end{eqnarray}

## # A tibble: 1,816 × 6
##    date       op2_dk2_price op2_dk2_load op2_dk2_wind op2_dk2_pen op2_dk2_exp
##    <date>             <dbl>        <dbl>        <dbl>       <dbl>       <dbl>
##  1 2010-01-01          3.70         7.45         4.69       -2.76      0.271 
##  2 2010-01-02          3.79         7.48         5.06       -2.43      0.0560
##  3 2010-01-03          3.76         7.52         4.15       -3.34      0.524 
##  4 2010-01-04          3.87         7.60         4.52       -3.09      0.679 
##  5 2010-01-05          3.86         7.61         4.37       -3.25      0.606 
##  6 2010-01-06          3.85         7.67         4.32       -3.34      0.441 
##  7 2010-01-07          4.19         7.67         3.98       -3.69      0.461 
##  8 2010-01-08          4.16         7.63         5.79       -1.83      0.964 
##  9 2010-01-09          3.82         7.52         6.37       -1.14      0.895 
## 10 2010-01-10          3.83         7.55         6.28       -1.27      0.961 
## # ℹ 1,806 more rows
## GUM log-variance equation:
## 
##              reg.no. keep       coef  std.error  t-stat   p-value    
## vconst             1    1  4.2344399  2.7765054  2.3259 0.1272349    
## arch1              2    0  0.1714824  0.0361237  4.7471 2.229e-06 ***
## arch2              3    0 -0.0597270  0.0381317 -1.5663 0.1174481    
## arch3              4    0  0.0078438  0.0267830  0.2929 0.7696591    
## arch4              5    0  0.0618698  0.0273163  2.2649 0.0236361 *  
## arch5              6    0 -0.0015604  0.0274335 -0.0569 0.9546475    
## arch6              7    0  0.1261577  0.0272541  4.6289 3.941e-06 ***
## arch7              8    0  0.0939292  0.0282651  3.3232 0.0009081 ***
## arch8              9    0  0.0070582  0.0250930  0.2813 0.7785286    
## arch9             10    0 -0.0520225  0.0250207 -2.0792 0.0377440 *  
## arch10            11    0  0.0097418  0.0250345  0.3891 0.6972235    
## arch11            12    0  0.0199281  0.0248816  0.8009 0.4232864    
## arch12            13    0 -0.0309765  0.0248653 -1.2458 0.2130127    
## arch13            14    0  0.0499864  0.0248789  2.0092 0.0446677 *  
## arch14            15    0  0.0469284  0.0236393  1.9852 0.0472771 *  
## logEqWMA(2)       16    0  0.3177781  0.0845986  3.7563 0.0001780 ***
## logEqWMA(5)       17    0  0.3188766  0.1460412  2.1835 0.0291315 *  
## logEqWMA(7)       18    0 -0.1314522  0.1587771 -0.8279 0.4078361    
## logEqWMA(14)      19    0 -0.1143233  0.0949523 -1.2040 0.2287466    
## op2_dk2_load      20    0 -1.4300365  3.2761761 -0.4365 0.6625301    
## op2_dk2_wind      21    0  0.9197847  3.2663561  0.2816 0.7782880    
## op2_dk2_pen       22    0 -0.8991733  3.2676477 -0.2752 0.7832141    
## op2_dk2_exp       23    0  0.3878186  0.0657336  5.8999 4.346e-09 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Diagnostics:
## 
##                      Chi-sq df p-value
## Ljung-Box AR(1)    481.1795  1  0.0000
## Ljung-Box ARCH(15)   3.1111 15  0.9995

## 17 path(s) to search
## Searching: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

## 
##   Path 1: 3 4 6 9 22 11 21 18 12 13 20 14 19 15 10 17 
##   Path 2: 4 6 9 22 11 21 12 19 13 14 20 3 18 15 10 17 
##   Path 3: 5 6 22 4 9 11 21 18 13 12 20 3 14 19 10 15 
##   Path 4: 6 22 9 4 11 21 12 19 13 14 20 3 18 15 10 17 
##   Path 5: 9 6 22 4 11 21 12 19 13 14 20 3 18 15 10 17 
##   Path 6: 10 11 4 9 6 22 18 21 12 13 20 3 14 19 15 17 
##   Path 7: 11 6 22 9 4 21 12 19 13 14 20 3 18 15 10 17 
##   Path 8: 12 6 9 22 4 11 21 19 13 14 20 3 18 15 10 17 
##   Path 9: 13 6 9 22 11 4 21 12 18 20 3 14 19 15 10 17 
##   Path 10: 14 6 9 22 4 11 21 19 12 13 20 3 18 15 10 17 
##   Path 11: 15 6 22 4 9 11 21 19 12 13 20 18 3 14 10 17 
##   Path 12: 17 9 22 11 21 6 4 12 13 20 19 18 14 3 10 15 
##   Path 13: 18 6 22 9 4 11 21 12 13 20 3 14 19 15 10 17 
##   Path 14: 19 6 9 11 22 4 21 12 13 14 20 3 18 15 10 17 
##   Path 15: 20 6 9 4 11 18 12 13 21 22 3 14 19 15 10 17 
##   Path 16: 21 6 9 4 11 22 12 19 13 14 20 3 18 15 10 17 
##   Path 17: 22 6 9 4 11 21 12 19 13 14 20 3 18 15 10 17 
## 
## Terminal models:
## 
##          info(sc)      logl    n k
## spec 1: 0.1894386 -144.4459 1802 7
## spec 2: 0.1446259 -104.0697 1802 7
## 
## Retained regressors (final model):
## 
##   vconst   arch1   arch6   arch7   logEqWMA(2)   logEqWMA(5)   op2_dk2_exp 
## % Date: Tue Sep 17 14:26:55 2024 
## % LaTeX code generated in R 4.3.1 by the gets package
## % Note: The {eqnarray} environment requires the {amsmath} package
## \begin{eqnarray}
##   \widehat{e} &=& \underset{(0.0318)}{0.3510} + \underset{(0.0236)}{0.2022}arch1 + \underset{(0.0230)}{0.1134}arch6 + \underset{(0.0551)}{0.0841}arch7 + \underset{(0.0557)}{0.1975}logEqWMA(2) + \underset{(0.0535)}{0.2471}logEqWMA(5) + \underset{(NA)}{0.3543}op2_dk2_exp - \underset{(NA)}{1.0021}Elnz2 \\[2mm] 
##    && R^2=NA \qquad \widehat{\sigma}=0.3241 \qquad LogL=-104.0697 \qquad T = 1802 \nonumber \\ 
## \end{eqnarray}

## # A tibble: 1,816 × 6
##    date       p_dk2_price p_dk2_load p_dk2_wind p_dk2_pen p_dk2_exp
##    <date>           <dbl>      <dbl>      <dbl>     <dbl>     <dbl>
##  1 2010-01-01        3.70       7.48       5.03     -2.43     0.223
##  2 2010-01-02        3.99       7.63       5.11     -2.51     0.156
##  3 2010-01-03        3.84       7.61       3.55     -4.05     0.389
##  4 2010-01-04        4.53       7.75       5.58     -2.17     0.532
##  5 2010-01-05        4.18       7.82       4.39     -3.43     0.317
##  6 2010-01-06        4.00       7.75       4.25     -3.50     0.374
##  7 2010-01-07        5.23       7.75       4.39     -3.36     0.528
##  8 2010-01-08        6.13       7.76       5.33     -2.43     0.761
##  9 2010-01-09        3.98       7.62       6.32     -1.30     0.955
## 10 2010-01-10        3.88       7.65       6.38     -1.27     0.839
## # ℹ 1,806 more rows
## GUM log-variance equation:
## 
##              reg.no. keep        coef   std.error  t-stat   p-value    
## vconst             1    1  0.82023915  2.40732128  0.1161  0.733309    
## arch1              2    0  0.16612433  0.03539240  4.6938 2.887e-06 ***
## arch2              3    0 -0.09098698  0.03695970 -2.4618  0.013919 *  
## arch3              4    0  0.01414347  0.02610639  0.5418  0.588050    
## arch4              5    0  0.05897363  0.02695908  2.1875  0.028834 *  
## arch5              6    0  0.02932308  0.02681526  1.0935  0.274313    
## arch6              7    0  0.07237627  0.02715552  2.6653  0.007763 ** 
## arch7              8    0  0.11145787  0.02772556  4.0200 6.062e-05 ***
## arch8              9    0  0.06486690  0.02457404  2.6397  0.008372 ** 
## arch9             10    0 -0.02695332  0.02483284 -1.0854  0.277896    
## arch10            11    0  0.00098512  0.02464778  0.0400  0.968123    
## arch11            12    0 -0.02547469  0.02467143 -1.0326  0.301951    
## arch12            13    0 -0.02409036  0.02456959 -0.9805  0.326975    
## arch13            14    0 -0.02925946  0.02462890 -1.1880  0.234987    
## arch14            15    0  0.12956578  0.02388867  5.4237 6.636e-08 ***
## logEqWMA(2)       16    0  0.26034008  0.08189343  3.1790  0.001503 ** 
## logEqWMA(5)       17    0 -0.25387549  0.14748055 -1.7214  0.085349 .  
## logEqWMA(7)       18    0  0.41198687  0.18593513  2.2158  0.026834 *  
## logEqWMA(14)      19    0 -0.09416325  0.13263956 -0.7099  0.477848    
## p_dk2_load        20    0 -2.31956939  4.51660852 -0.5136  0.607620    
## p_dk2_wind        21    0  2.25826603  4.48580358  0.5034  0.614728    
## p_dk2_pen         22    0 -2.23034813  4.48540797 -0.4972  0.619078    
## p_dk2_exp         23    0  0.34339376  0.06962025  4.9324 8.880e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Diagnostics:
## 
##                     Chi-sq df p-value
## Ljung-Box AR(1)    433.832  1 0.00000
## Ljung-Box ARCH(15)  17.746 15 0.27626

## 18 path(s) to search
## Searching: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

## 
##   Path 1: 3 11 4 22 20 19 21 6 13 10 17 16 5 14 12 9 7 
##   Path 2: 4 11 22 20 21 19 6 13 10 17 14 5 12 9 18 3 
##   Path 3: 5 11 22 20 4 12 21 6 17 14 10 13 19 9 18 3 
##   Path 4: 6 11 4 22 20 21 13 10 12 14 17 5 19 9 18 3 
##   Path 5: 7 11 22 20 4 13 21 10 14 12 6 5 17 16 3 19 9 
##   Path 6: 9 11 19 22 20 4 10 21 6 13 17 14 18 12 5 3 
##   Path 7: 10 11 4 22 20 21 13 6 12 14 17 5 19 9 18 3 
##   Path 8: 11 22 20 4 21 19 6 13 10 17 14 5 12 9 18 3 
##   Path 9: 12 11 4 22 20 21 6 14 10 13 17 5 19 9 18 3 
##   Path 10: 13 11 22 20 4 21 6 10 12 14 17 5 19 9 18 3 
##   Path 11: 14 11 22 20 4 21 6 12 10 13 17 5 19 9 18 3 
##   Path 12: 16 11 4 3 22 20 6 21 10 13 14 17 5 12 19 9 7 
##   Path 13: 17 11 4 22 20 6 21 19 13 10 14 5 12 9 18 3 
##   Path 14: 18 11 17 4 22 20 21 19 6 13 10 14 12 9 5 3 
##   Path 15: 19 11 22 20 4 21 6 13 10 17 14 5 12 9 18 3 
##   Path 16: 20 11 21 4 22 19 6 13 10 17 14 5 12 9 18 3 
##   Path 17: 21 11 20 4 22 19 6 13 10 17 14 5 12 9 18 3 
##   Path 18: 22 11 20 4 21 19 6 13 10 17 14 5 12 9 18 3 
## 
## Terminal models:
## 
##          info(sc)      logl    n k
## spec 1: 0.4913591 -420.2246 1802 6
## spec 2: 0.5227186 -444.7312 1802 7
## 
## Retained regressors (final model):
## 
##   vconst   arch1   arch7   arch14   logEqWMA(7)   p_dk2_exp 
## % Date: Tue Sep 17 14:26:55 2024 
## % LaTeX code generated in R 4.3.1 by the gets package
## % Note: The {eqnarray} environment requires the {amsmath} package
## \begin{eqnarray}
##   \widehat{e} &=& \underset{(0.0233)}{0.3995} + \underset{(0.0241)}{0.2392}arch1 + \underset{(0.0219)}{0.1411}arch7 + \underset{(0.0542)}{0.1297}arch14 + \underset{(0.0666)}{0.3093}logEqWMA(7) + \underset{(NA)}{0.3586}p_dk2_exp - \underset{(NA)}{1.1591}Elnz2 \\[2mm] 
##    && R^2=NA \qquad \widehat{\sigma}=0.3335 \qquad LogL=-420.2246 \qquad T = 1802 \nonumber \\ 
## \end{eqnarray}

Extending to Europe

Hourly prices

Using data from EMBER on European wholesale electricity prices (hourly - daily and monthly also exist), I perform the same analysis as above. The hourly prices are the day-ahead wholesale prices by country.

Spotlight on UK, Germany, Denmark

Here, see the time series for Germany and Denmark (to compare to the data avialable in Rintamaki et al. - color portion of the TS plots). UK spotlight to discuss potential interest to David.

Daily and Monthly Prices